\section{Introduction}
Inertial particles moving in a turbulent flow do not simply trace the paths of fluid elements since particle inertia allows the particles to detach from the local fluid. This causes small-scale spatial clustering even
in incompressible turbulence~\citep{Max87,Dun05,Bec07,Vas08,Bra14}.
Moreover, {\it preferential sampling} of  regions with high vorticity/strain is  observed for particles that are lighter/heavier than the fluid.
As a result, the statistical properties of inertial heavy or light particles
 [for example water droplets in turbulent clouds~\citep{Sha03} or air bubbles in water~\citep{Maz03,Ren05}] in turbulence are highly non-trivial both at sub-viscous and larger scales.
An important  example is the case of  inertial particle accelerations \citep{Bec06c,Qur08,Gib10,Gib12,Vol11,Pra12}, see also
the review by \cite{TB2009}. At large inertia, singularities (caustics) occur
in the dynamics of heavy particles \citep{Fal02,Wil05}, giving rise
to large relative velocities between close  particles \citep{Sun97,Fal07c,Bec10,Gus11b,Gus13c},
and leading eventually to an enhancement of collision rates~\citep{Wil06,Bec14}.\\
The dynamics of light particles is important because of its relevance to  fundamental and applied questions. Light particles
can be used as small probes that  preferentially track high vorticity structures, highlighting statistical
 and topological properties of the underlying  fluid conditioned on those structures.  In the limit of high volume fractions
they might also have a complex feedback on the flow, including the case of reducing the turbulent drag \citep{Jac10,Ber05,Ber07}.
Compared to heavy particles the dynamics of light particles is more difficult to analyse because pressure-gradient
forces~\citep{Tch47} and  added-mass effects must be taken into account.
Apart from the fact that light particles tend to be drawn into vortices there is little theoretical analysis of their dynamics in turbulent flows.

This motivated us to formulate a {\it closure scheme} that allows us to compute inertial accelerations of both heavy and light particles in turbulence. We describe the scheme and  we test it by comparing its predictions to results of direct numerical simulations (DNS).
We consider a simple model for the particle dynamics, taking into account added-mass and pressure-gradient forces, but neglecting the effect of buoyancy on the acceleration as well as the Basset-Boussinesq history force, as in many previous DNS studies \citep{Bab00,Bec05,Cal08,Cal09,Vol08}. We comment on these questions in the conclusions.

Our scheme for the inertial particle accelerations approximates the particle paths by the Lagrangian fluid-element paths.
It is a closure for the particle equation of motion that neglects {\it inertial preferential sampling} such as heavy inertial particles being centrifuged out of long-lived vortical regions of the flow~\citep{Max87}, so that they preferentially sample straining regions, or light particles which by contrast are drawn into vortices \citep{Cal08,Cal08b}.
The latter is simply a consequence of the fact that
light particles are influenced by pressure gradients in the undisturbed flow.
Our approximation cannot be quantitatively correct when inertial preferential sampling is significant, but overall it yields a good qualitative description of
how particle accelerations depend on the particle density relative to that of the fluid, and upon the {\it Stokes number}, a dimensionless measure of particle inertia.
Moreover, and more interestingly, the closure also predicts highly non-trivial properties of the intermittent, non-Gaussian acceleration fluctuations, as the magnitude of inertial effects and the  Reynolds number of the turbulence change.  For example, the closure scheme predicts that the flatness of the acceleration  develops a  maximum or a minimum as a function of $\st$ for light or heavy particles. This is in good qualitative agreement with the measurements on the DNS data.
For heavy particles the closure  fails when  inertial preferential sampling affects inertial particle accelerations, i.e.  for Stokes numbers of order unity,
see Fig.~1{\bf b} in \citep{Bec06c}. In fact for heavy particles our closure is equivalent to the \lq filtering\rq{} mechanism
discussed by \cite{Bec06c}. For light particles, by contrast, comparisons to DNS results show
that  inertial preferential sampling effects are strong only at very  large Stokes numbers, $\st \sim 10$,
leading to an enlarged range of small and intermediate values of $\st$ where the closure works qualitatively well.

The remainder of this article is organized as follows. In the next section we formulate the problem,
introduce the equations of motion and qualitatively discuss inertial  preferential sampling.
 In Section \ref{sec:closure} we
describe our Lagrangian closure scheme for inertial particle accelerations.
 Data from DNS of turbulent flows are analysed
in Section~\ref{sec:turbo} together with a detailed comparison to the predictions from the closure scheme.
In Section~\ref{sec:stat_model},
 we assess the potentialities of the closure on a data set obtained using a stochastic surrogate for the fluid velocity field.
Section~\ref{sec:conclusions}
contains the  conclusions.

\section{Formulation of the problem}
\label{sec:problem}
Many studies have considered the dynamics of heavy inertial particles, much denser than the carrying fluid.
When the particles are very heavy and at the same time very small (point particles)
the motion is simply determined by Stokes' drag.
The dynamics of light particles by contrast is also affected by pressure gradients of the unperturbed fluid,
and added-mass effects. Neglecting the effect of gravitational settling (and thus buoyancy) the equation of motion reads:
\begin{equation}
\begin{cases}
    \dot{\ve r}_t = {\ve v }_t\,,\\
    \dot{\ve v }_t = \beta  {\rm D}_t \ve u(\ve r_t,t) + \big(\ve u(\ve r_t,t)  - {\ve v}_t\big)\big/\tau_{\rm s}\,,
    \label{mrfc}
\end{cases}
\end{equation}
where $ \ve r_t$ is the  particle position at time $t$, $ \ve v_t$, is the particle velocity,
$\ve u(\ve r_t, t)$ is the velocity field of the undisturbed fluid and
${\rm D}_t\ve u = \partial_t \ve u + (\ve u\cdot\ve \nabla)\ve u$ is the Lagrangian derivative.
The dimensionless constant $\beta= 3\rho_{\rm f}/(\rho_{\rm f} + 2 \rho_{\rm p})$ accounts for the contrast between
particle density $\rho_{\rm p}$ and fluid density $\rho_{\rm f}$, while the Stokes number is defined as $\st = \tau_{\rm s}/\tau_{\rm K}$
where the particle response time is $\tau_{\rm s} = R^2 /(3 \nu \beta)$, $R$ is the particle radius,
$\nu$ the kinematic viscosity of the flow and $\tau_{\rm K} =\sqrt{\nu/\epsilon}$ the Kolmogorov time
defined in terms of the fluid energy dissipation, $\epsilon$. Many studies have employed
this model \citep{Bab00,Bec05,Cal08,Cal09,Vol08}.
The model takes into account added-mass effects but neglects buoyancy forces. It is an open question under which circumstances this is a quantitative model for the acceleration of small particles in turbulence.

The following analysis is based on Eqs.~(\ref{mrfc}).
It is important to first understand this case before  addressing more realistic situations (inclusion of buoyancy, finite size, collisions and feedback on the flow).
The sources of the difficulties are twofold. First,
even in the much simpler case of a non-turbulent  Eulerian velocity field $\ve u(\ve r,t)$
the particle dynamics is still complicated and often chaotic, simply because Eqs.~(\ref{mrfc}) are nonlinear.
Second, turbulence makes the problem even harder due to the existence of  substantial
spatial and temporal fluctuations. This results in chaotic Lagrangian dynamics of fluid elements.
Note that even though Lagrangian fluid elements sample space uniformly,
their instantaneous motion is in general correlated with structures in the underlying flow.
This implies that multi-time Lagrangian flow correlation functions evaluated along tracer trajectories
do not coincide with the underlying Eulerian correlation functions which are evaluated at fixed positions in space.
Particles that are heavier or lighter than the fluid may detach from the flow if they have inertia.
This leads to the {\it inertial preferential sampling} mentioned in the introduction. As a result, the flow statistics
experienced by an inertial particle differs from that of a Lagrangian fluid element.

\section{Lagrangian closure}
\label{sec:closure}
Let us notice that the dynamics determined by  equation (\ref{mrfc})
tends to the evolution of a tracer in both limits $\st \rightarrow 0$ and $\beta \rightarrow 1$. Indeed,
in the limit $\tau_s \rightarrow 0$, imposing  a finite Stokes drag leads to $\ve v_t = \ve u + O(\tau_s)$. For  $\beta = 1$, by contrast
we may approximate the material derivative along fluid tracers trajectories with the derivative along the trajectory of a particle, ${\rm D}_t \ve u \sim \dot{\ve u}$, and  consistently check that  the evolution of (\ref{mrfc}) leads to an exponential relaxation of the evolution of the particle to the trajectory of the tracer.
Hence, the idea is to approximate the effects of inertial forces on the particle trajectory
starting from the evolution of tracers. This approach cannot be exact.
It is for example known that vortices are preferentially sampled by inertial particles.
Nevertheless, it is important to understand and quantify how big the difference is as a function of the distance in the parameter phase space, $(\st, \beta)$, from the two lines $\st=0$ or $\beta=1$ where the closure must be exact (see. Fig. \ref{fig:phasespace}).
The starting point is to evaluate (\ref{mrfc}) along tracer trajectories:
\begin{equation}
\begin{cases}
    \dot{\ve r}^{({\rm L})}_t = \ve u(\ve r^{({\rm L})}_t,t)\,, \\
    \dot{\ve v_t} = \beta  {\rm D}_t   \ve u(\ve r^{({\rm L})}_t,t) + \big( \ve u(\ve r^{({\rm L})}_t,t)  - {\ve v_t}\big)\big/\tau_{\rm s}\,,
    \label{mrfc2}
\end{cases}
\end{equation}
where $\ve r^{({\rm L})}_t$ denotes the Lagrangian trajectory of a tracer particle.
This approximation is a {\it closure} in the sense that the first equation in (\ref{mrfc2}) is independent of the second one,
and the second equation can be solved in terms of the Lagrangian velocity statistics $\ve u(\ve r^{({\rm L})}_t,t)$ of the underlying flow.
We solve (\ref{mrfc2}) for $\ve v_t$ (disregarding initial conditions here and below because these do not matter for the
 steady-state statistics) to obtain
\begin{align}
\ve v_t&=\beta\ve u(\ve r_{t}^{(\rm L)},t)+\frac{1-\beta}{\tau_{\rm s}}\int_0^t{\rm d}t_1e^{(t_1-t)/\tau_{\rm s}}\ve u(\ve r_{t_1}^{(\rm L)},t_1)\,.
\eqnlab{v_implicit}
\end{align}
Using this expression, the particle acceleration follows from the second of Eqs.~(\ref{mrfc2})
\begin{align}
\ve a_t&=\beta\frac{{\rm D}\ve u}{{\rm D}t}(\ve r_{t}^{(\rm L)},t)+\frac{1-\beta}{\tau_{\rm s}}\int_0^t{\rm d}t_1e^{(t_1-t)/\tau_{\rm s}}\frac{{\rm D}\ve u}{{\rm D}t}(\ve r_{t_1}^{(\rm L)},t_1)\,.
\eqnlab{a_implicit}
\end{align}
Using \Eqnref{a_implicit} we express two-time acceleration statistics in terms of the two-point Lagrangian acceleration correlation function
\begin{equation}
C_{\rm L}(t) \equiv
\langle {\rm D}_t\ve u(\ve r_t^{(\rm L)},t)\cdot{\rm D}_t\ve u(\ve r_0^{(\rm L)},0)\rangle\,.
\end{equation}
In the steady-state limit we find
\begin{align}
\langle \ve a_t\cdot\ve a_0\rangle
&= \beta^2C_{\rm L}(t)+\frac{1-\beta^2}{\tau_{\rm s}}\left[
\cosh\Big[\frac{t}{\tau_{\rm s}}\right]\int_t^\infty{\rm d}t_1e^{-t_1/\tau_{\rm s}}C_{\rm L}(t_1)\nonumber\\
&\hspace*{4cm}
+e^{-t/\tau_{\rm s}}\int_0^t{\rm d}t_1 \cosh\left[\frac{t_1}{\tau_{\rm s}}\right]C_{\rm L}(t_1)
\Big]\,.
\label{aCorr}
\end{align}
The acceleration variance is obtained by letting $t\to 0$ in Eq.~(\ref{aCorr})
\begin{equation}
\langle\ve a^2\rangle=\beta^2C_{\rm L}(0)+\frac{1-\beta^2}{\tau_{\rm s}}\int_0^\infty{\rm d}t_1e^{-t_1/\tau_{\rm s}}C_{\rm L}(t_1)\,.
\label{aVar}
\end{equation}
Similarly, the fourth moment of the particle acceleration is obtained as:
\begin{align}
\langle |\ve a|^4\rangle&
= \beta^4C_{\rm L}(0,0,0)
-4\frac{(\beta-1)\beta^3}{\tau_{\rm s}}\int_0^\infty{\rm d}t_1e^{-t_1/\tau_{\rm s}}C_{\rm L}(-t_1,0,0)\nn\\
& +6\frac{(\beta-1)^2\beta^2}{\tau_{\rm s}^2}\int_0^\infty{\rm d}t_1\int_0^\infty{\rm d}t_2e^{-(t_1+t_2)/\tau_{\rm s}}C_{\rm L}(-t_1,-t_2,0) \nn\\
& -\frac{(\beta-1)^3(3\beta+1)}{\tau_{\rm s}^3}\int_0^\infty{\rm d}t_1\int_0^\infty{\rm d}t_2\int_0^\infty{\rm d}t_3e^{-(t_1+t_2+t_3)/\tau_{\rm s}}C_{\rm L}(-t_1,-t_2,-t_3)\,.
\label{aQuad}
\end{align}
Here isotropy of the acceleration components $\langle a_ia_ja_ka_l\rangle=[\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}]\langle a_1^4\rangle/3$ was used to express $\langle |\ve a|^4\rangle$ in terms of the four-point Lagrangian correlation function
\begin{align}
C_{\rm L}(t_1,t_2,t_3)\equiv \frac{d(d+2)}{3}\langle {\rm D}_tu_1(\ve r_{t_1}^{(\rm L)},t_1){\rm D}_tu_1(\ve r_{t_2}^{(\rm L)},t_2){\rm D}_tu_1(\ve r_{t_3}^{(\rm L)},t_3){\rm D}_tu_1(\ve r_{0}^{(\rm L)},0)\rangle\,,
\end{align}
where ${\rm D}_tu_1$ is a component of the fluid acceleration and $d$ is the spatial dimension.
Eqs.~(\ref{aCorr}), (\ref{aVar}) and (\ref{aQuad}) express the fluctuations of inertial particle accelerations in terms of Lagrangian correlation functions
of the underlying flow, and predict how the inertial particle accelerations depend on $\beta$ and $\st$.
The integrals in Eqs.~(\ref{aVar}) and (\ref{aQuad}) are hard to evaluate numerically for very small and for very large values of $\st$.
When $\st$ is small, the exponential factors in the right-hand side of (\ref{aVar}) and (\ref{aQuad}) become singular and one needs a
 very high sampling frequency of the fluid acceleration along the particle trajectory to evaluate the integrals reliably.
When $\st$ is large, on the other hand, the integrals sum up large-time contributions with large fluctuations due to the
finite length of experimental and numerical  trajectories.
Before we proceed to a quantitative assessment of the model let us make a few general remarks about the range of applicability and accuracy of the approximation made. First, in Eqs. (\ref{aVar}) and (\ref{aQuad}) we need to evaluate the acceleration correlation function of the tracers for general values of $t$. This might be seen as a problem because the trajectories of tracers and of inertial particles must depart on a time scale of the order of the Lyapunov time. On the other hand, the acceleration correlation function of the tracer is known to decay on a time of the order of the Kolmogorov time, $\tau_K$, which is ten times smaller than the typical Lyapunov time of heavy particles \citep{Bec06} and stays close to zero in the inertial range of scales \citep{Falk12}. Therefore we do not see any problem in the evaluation of the integrals in the above equations. Second,  we are only interested in stationary properties of the inertial particle statistics and we always assume that the initial conditions for all particles are chosen from their stationary distribution functions. As a result no influence of the initial condition should appear in the closure.
The Lagrangian closure adopted here can also be applied to correlation functions of other observables of the particle, or of the flow, provided that the underlying Lagrangian correlation functions decay quickly enough.
Here,  we focus  on the acceleration statistics because of their highly non-Gaussian properties and
high sensitivity to the parameters $\beta$ and $\st$. Finally, let us stress that our closure is fully based on Lagrangian properties and does require the use of any Eulerian correlation function, as opposed to closures based on the fast Eulerian approach by \cite{Fer01} or the  Lagrangian-Eulerian  closure
to predict two-particle distributions \citep{Zai03,Ali06,Der00,Pan10}.
To make further progress we need
to determine the Lagrangian correlation functions.
In Section \ref{sec:turbo} we present the most important new results, we determine the Lagrangian correlation functions by DNS of inertial particle dynamics in turbulence, substitute into Eqs.~(\ref{aCorr}), (\ref{aVar}) and (\ref{aQuad}) and assess the accuracy of these equations in predicting particle acceleration fluctuations and correlations.

\section{Direct numerical simulations}
\label{sec:turbo}
\subsection{Simulation method}
We present here the analysis of data obtained
from  DNS of a homogeneous and isotropic turbulent flow seeded with point-like particles with different inertia and
particle-fluid density
ratios  (see Fig.~\ref{fig:phasespace} for a summary of the $(\st,\beta)$ values available).
\begin{figure}
\hspace*{-0.05in}
\includegraphics[width=13.5cm]{Fig1.eps}
\vspace{-0.2cm}
\caption{\small {\em (Online color).}
Parameter-space showing available DNS data sets with $\re_\lambda=185$ and $\re_\lambda=400$.
For values of $\st$ smaller than $0.05$ the $x$ axis is linear, while it is logarithmic for values of $\st$ larger than $0.05$.
For each data set with $\re_\lambda=185$ we have analysed a total of $130000$ trajectories of duration $6T_L$ and for each data set with $\re_\lambda=400$ we have analysed a total of $200000$ trajectories of duration $2.5T_L$.
Level curves $\st(1-\beta)=\epsilon$ for constants $\epsilon=\{-1,-0.1,-0.01,0,0.01,0.1,1\}$ are plotted as black lines.
Parameter families:
With $\re_\lambda=185$:
$\beta=0$ (red,$\circ$),
$\beta=0.25$ (magenta,$\vartriangle$),
$\beta=0.5$ (cyan,$\triangledown$),
$\beta=0.75$ (green,$\triangleright$),
$\beta=1$ (black,$\Box$),
$\beta=1.25$ (brown,$\Diamond$),
$\beta=1.5$ (purple,$\triangledown$),
$\beta=2$ (orange,$\triangleright$),
$\beta=2.5$ (dark green,$\triangleleft$),
$\beta=3$ (blue,$\vartriangle$).
With $\re_\lambda=400$:
$\beta=0$ (red,\marker{6}).
Additional data from~\cite{Cal09,Pra12}:
$\beta=3$ (blue,\marker{7}).
}
\label{fig:phasespace}
\end{figure}
The data set was previously obtained by \citep{becJFM2010}. The flow obeys the  Navier-Stokes equations for
an incompressible velocity field ${\ve u}(\ve x,t)$:
\begin{equation}
  \partial_t\ve u + \ve u \cdot \nabla \ve u = -\nabla p +
  \nu\nabla^2\ve u +\ve f,\quad \nabla\cdot\ve u = 0\,.
  \label{eq:ns}
\end{equation}
The external forcing $\ve f$ is statistically homogeneous, stationary and isotropic,
injecting energy in the first low wavenumber shells, by keeping their spectral content constant~\citep{She}.
The viscosity $\nu$ is set such that the Kolmogorov
length scale $\eta\approx \delta x$, where $\delta x$ is the grid
spacing. The numerical domain is  $2\pi$-periodic
in the three directions. We use a fully dealiased
pseudospectral algorithm with second-order Adam-Bashforth
time stepping. For details see \citep{Bec06c,Bec06d}. Two series of DNS are analysed:
 Run I, with a numerical resolution of $512^3$ grid
points, and the Reynolds number at the Taylor scale $\re_\lambda
\approx 200$; Run II, with $2048^3$ resolution and $\re_\lambda \approx
400$. Details can be found in Table~\ref{table}.
\begin{table*}
  \centering

 \caption{\label{table} Eulerian parameters for the two runs analyzed
    in this Article: Run I and Run II in the text. $N$ is the number of grid
    points in each spatial direction; $\re_{\lambda}$ is the
    Taylor-scale Reynolds number; $\eta$ is the Kolmogorov dissipative
    scale; $\delta x=\mathcal{L}/N$ is the grid spacing, with
    $\mathcal{L}=2\pi$ denoting the physical size of the numerical
    domain; $\tau_{\rm K}=\left( \nu/\varepsilon \right)^{1/2}$ is the
    Kolmogorov dissipative time scale; $\varepsilon$ is the average
    rate of energy injection; $\nu$ is the kinematic viscosity;
    $t_{\mathrm{dump}}$ is the time interval between two successive
    dumps along particle trajectories; $\delta t$ is the time step;
    $T_L=L/U_0$ is the eddy turnover time at the integral scale $L=\pi$,
    and $U_0$ is the typical large-scale velocity.}
\mbox{}\\
  \begin{tabular}{ccccccccccc}
    \hline\\[-7pt]
    & $N$ & $\re_{\lambda}$ & $\eta$ & $\delta x$ & $\varepsilon$ &
    $\nu$ & $\tau_{\rm K}$ & $t_{\mathrm{dump}}$ & $\delta t$ & $T_L$
    \\[+2pt]\hline\\[-7pt] Run I& 512 & 185 & 0.01 & 0.012 & 0.9 & 0.002 &
    0.047 & 0.004 & 0.0004 & 2.2 \\ Run II& 2048 & 400 & 0.0026 & 0.003 &
    0.88 & 0.00035 & 0.02 & 0.00115 & 0.000115 & 2.2\\[+2pt]\hline
  \end{tabular}
\end{table*}

\subsection{Comparison for acceleration variance and flatness against DNS}
\label{sec:dns_results}
In Fig.~\ref{fig:accvarDNSotherbetas}{\bf a} we show the comparison of the acceleration variance, $\langle a^2 \rangle$,
 to DNS data as a function of $\st$
for different values of  $\beta$.
Consider first heavy particles ($ \beta <1 $).  It is clear that the closure (\ref{mrfc2}) captures the general trend and it
becomes better and better for larger Stokes numbers. Similarly, this approximation must become exact as $\st \to 0$, but
 the DNS data set does not contain values of $\st$ small enough to reach this limit.
At intermediate Stokes numbers the  Lagrangian closure described in  Section \ref{sec:closure} does not
match the DNS results.  This mismatch for Stokes numbers between $0.1$ and $1$ is certainly
 due to preferential sampling, as already remarked  by \cite{Bec06c}. Yet, the closure predicts a small Reynolds dependency (compare solid and dashed red lines for $\beta=0$)
inherited  by the dependency of the fluid tracers. Such  a small variation is not detectable within
 the accuracy of our numerical data.
Now consider light particles ($\beta >1$). Fig. ~\ref{fig:accvarDNSotherbetas}{\bf a} shows  that also in this case the agreement between the
 DNS data and the closure scheme is good,  except around $\st \sim 10$ where the closure underestimates the DNS data.
\begin{figure}
\hspace*{-0.05in}
\includegraphics[width=13.5cm]{Fig2.eps}
\vspace{-0.2cm}
\caption{\small {\em (Online colour).}
Acceleration variance ({\bf a}) and flatness ({\bf b}) for DNS data at changing $\beta$, $\st$ and Re.
{\bf a}: points correspond to the DNS data (same symbols of Fig.\ref{fig:phasespace}). Solid lines (labeled with their corresponding value of $\beta$) show the closure scheme prediction for the acceleration variance of light and heavy particles, Eq.~(\ref{aVar}), normalized with the fluid variance, for all data from  RUN I;
dashed line corresponds to the closure for RUN II.
{\bf b}: the flatness measured on  the DNS data and the one
predicted by the closure scheme, Eqs.~(\ref{aQuad}), for heavy and light particles.
Thin dashed black line shows the limit of normal distributed acceleration components.
Additional data for $\beta=3$ (blue,\protect\includegraphics[width=2mm,clip]{markBW7.eps}) is omitted in panel {\bf b} because the flatness was not evaluated in Refs.~\cite{Cal09,Pra12}.
}
 \figlab{accvarDNSotherbetas}
\end{figure}
In this case inertial preferential sampling must be  important. The reason
is that light particles are drawn into vortex filaments where they experience high accelerations.
Nevertheless, inertial preferential sampling must  become irrelevant in the limit $\st\to\infty$ as shown by the trend for very large $\st$ in the same figure.
It is interesting to
 remark that  the closure scheme  (solid line) becomes better and better the closer $\beta$ is to unity,  and/or the
smaller the Stokes number is, suggesting the possibility to develop a systematic perturbative expansion in the small parameter
$\epsilon = \st (1-\beta)$.
 Furthermore it is important to note that while the acceleration variance increases  monotonically as both $\beta$ and $\st$ increase,
the flatness of the acceleration, defined as
$$F_{\st,\beta} = \langle |\ve a|^4\rangle/\langle \ve a^2\rangle^2,$$ has a non-monotonic dependency on $\st$.
In  Fig.~\ref{fig:accvarDNSotherbetas}{\bf b} we show that light particles have a maximum in their flatness at $\st \sim 0.5$
 and heavy particles have minimum flatness for $\st>1$ (except for the case of very heavy particles with $\beta=0$).
Importantly, our closure approximation predicts these extrema qualitatively,
indicating that the  non-Gaussian tails observed numerically and experimentally \citep{Vol08} in the acceleration probability distribution function  of bubbles
 are not only due to  inertial preferential sampling, which is neglected by the closure.
Let us also note that the non-monotonicity shown by the model for $\st \sim 0.1$ and $\beta=0$, $0.25$ and $3$ are artefacts due to insufficient accuracy in the numerical evaluation of the integral in Eq.~(\ref{aQuad}).
In conclusion, remarkably, the closure approximation is in reasonable agreement with the DNS data, with the exception of those values of Stokes number where preferential
 sampling is important. This is an important result, supporting the idea that many key properties, including deviations from Gaussian statistics,
 of the acceleration statistics of inertial particles are due to the kinematic structure of the equation of
motion together with the non-trivial Lagrangian properties of the
fluid tracers evolution, and not only due to inertial preferential sampling.

\begin{figure}
\hspace*{-0.05in}
\includegraphics[width=13.5cm]{Fig3.eps}
\vspace{-0.2cm}
\caption{\small {\em (Online colour).}
Relative error $\Delta a^2$ \eqnref{relerr} between the DNS data and the closure scheme prediction for heavy particles ({\bf a}) and light particles ({\bf b}). Markers according to \Figref{phasespace}. Lines are drawn as a guide to the  eye.}
\figlab{acc_error_DNS}
\end{figure}
In order to quantify the accuracy of the closure described in \Secref{closure} we plot in Fig.~\ref{fig:acc_error_DNS} the relative error in the prediction of the acceleration variance
for both light and heavy particles:
\begin{equation}
\Delta a^2 = \left|1 - \frac{\langle \ve a^2 \rangle}{\langle \ve a^2 \rangle_{\rm DNS}}\right|\,,
\eqnlab{relerr}
\end{equation}
where with $\langle \ve a^2 \rangle$ we refer to the expression (\ref{aVar}) and with
$ \langle \ve a^2 \rangle_{\rm DNS}$ to the numerical value obtained from the DNS results. As one can see the approximation is never very bad, with a maximum discrepancy of the order of $30-40\%$ at those values where inertial preferential sampling is important, i.e. for $\st \sim O(1)$ for heavy particles and
for $\st \sim O(10)$ for light particles.

\section{Statistical Eulerian velocity model}
\label{sec:stat_model}
Mathematical analysis and numerical studies of the particle dynamics become easier when the turbulent
fluctuations of $\ve u(\ve r,t)$ are approximated by a stochastic process.
Following \cite{Gus15} we use a smooth, homogeneous and isotropic
Gaussian random velocity field with root-mean-squared speed $u_0$ and typical length and
time scales $\eta$ and $\tau$.
The model is characterized by a dimensionless number, the Kubo number, $\ku= \tau/(\eta/u_0)$,
that measures the degree of persistence of flow structures in time. Very small Kubo numbers correspond
to a rapidly fluctuating fluid velocity field. In this limit the closure approximation
described in \Secref{closure} is exact, inertial preferential sampling is negligible and
the Lagrangian correlation functions of tracer particles are well approximated by the Eulerian correlation functions.
In this limit it is also possible to perform a systematic perturbative expansion \citep{Gus15}. In this paper
we are  interested in comparing the validity of the Lagrangian closure for the statistical model at $\ku \sim O(1)$, where no analytical results can be obtained and to further compare them with the DNS results shown in \Secref{turbo}. The motivation  is the following. The statistical model has no 'internal intermittency', i.e. there is no Reynolds number dependency on the acceleration statistics (there is not even  the meaning of a Reynolds number). Nevertheless, once the Gaussian Eulerian velocity field is prescribed, we can calculate the acceleration probability density function of the fluid tracers. It turns out that this is not Gaussian and that it depends on the Kubo number,  due to the effect of the quadratic advection term, $\ve a_{\rm f}=\partial_t\ve u+\ku[\ve\nabla\ve u^{\rm T}]\ve u$. As a result, we expect that many of the properties shown by the acceleration distribution of
inertial particles evolved in real turbulent flows are shared by particles evolved in a Gaussian random flow.
Finally, a comparison between DNS data and the statistical model will allow us to assess further the importance of internal intermittency.
\subsection{Construction of the random velocity field}
For simplicity we discuss only the two-dimensional case. Generalization to three dimensions is straightforward.
The velocity field is given in terms of the streamfunction: $\ve u(\ve r,t)=\ve\nabla\psi(\ve r,t)\wedge\hat{\ve e}_3\,$, which is defined
as a superposition of Fourier modes with a Gaussian cutoff,
\begin{equation}
\label{eq:psi}
\psi(\ve x,t)=\frac{\eta^2u_0}{\sqrt{\pi}L}\sum_{\sve k}a_{\sve k}(t)e^{{\rm i}\sve x\cdot\sve k-k^2\eta^2/4}\,.
\end{equation}
Here the system size $L$ is put to $10\eta$, $k_i=2\pi n_i/L$ and $n_i$ are integers with an upper cutoff $|n_i|\le 2L/\eta$ because higher-order Fourier modes are negligible.
The resulting spatial correlation function of $\psi$ is Gaussian, if $L\gg\eta$ we have:
$\langle\psi(\ve x,0)\psi(\ve 0,0)\rangle=(u_0^2\eta^2/2)\,e^{-x^2/(2\eta^2)}.$
The random coefficients $a_{\sve k}(t)$ in $\psi$ are drawn from random Gaussian distributions with zero means, smoothly
correlated in time. To do that we used  an Ornstein-Uhlenbeck process  convolved  with a Gaussian kernel of the form
$
w(t)\equiv \exp[-t^2/(2t_0^2)]/(t_0\sqrt{2\pi})\,$ to have a smooth correlation function also for the acceleration.
The parameter $t_0$ must be small, $t_0\ll \tau$, in order for the flow field to
decorrelate at long times in a similar fashion as in fully developed turbulence.
The
Eulerian autocorrelation function of ${\ve u}$ is:
\begin{equation}
    \langle {\ve u}(\ve x_0,t)\cdot {\ve u}(\ve x_0,0)\rangle =
    \frac{u_0^2}{2\sqrt{\pi}}e^{-t^2/(4t_0^2)} \left(\E\left[\frac{t_0}{\tau} + \frac{t}{2t_0}\right]
    \right.
    + \left. \E\left[\frac{t_0}{\tau}-\frac{t}{2t_0}\right]\right)\,,
    \eqnlab{uCorrSmoothMain}
\end{equation}
where $\E(x) = \sqrt{\pi}\exp(x^2)\erfc(x)$.
Note that the flow field is homogeneous in space and also in time and
  $\langle{\ve u}(\ve x_0,t)\cdot{\ve u}(\ve x_0,0)\rangle$
is a function of $|t|$ only. For $ |t| \ll t_0$ this correlation
function is Gaussian and for $ |t| \gg t_0$ the correlation
function is exponential:
$\langle{\ve u}(\ve x_0,t)\cdot{\ve u}(\ve x_0,0)\rangle\sim e^{-|t|/\tau}$.
\subsection{Results for the random velocity model}
\label{sec:res_stat_model}
\begin{figure}
\hspace*{-0.05in}
\includegraphics[width=13.7cm]{Fig4.eps}
\vspace{-0.2cm}
\caption{
\small {\em (Online colour).}
Same as in Fig. \ref{fig:accvarDNSotherbetas} but for the statistical model given by Eq. (\ref{eq:psi}) at $\ku=5$.
}
\figlab{acc_var_stat}
\end{figure}
We consider first the acceleration variance. Simulation results for the statistical model are compared with the Lagrangian closure (\ref{aVar}-\ref{aQuad}) in Fig.~\ref{fig:acc_var_stat} by changing both $\st, \beta$  for $\ku=5$  (panel {\bf a}).
We observe a good agreement between the Lagrangian closure and the numerical simulations, comparable to what was observed for the
DNS data.
In \Figref{acc_var_stat}{\bf b} we show the results for the flatness. It is important to stress two facts. Also here, both  heavy and light particles depart from the corresponding fluid value with a qualitative trend similar to that observed for the DNS case in the previous section. Also for the random velocity field, the Lagrangian closure works qualitatively well. The departure from the numerical data is a signature of the corresponding importance of preferential sampling at those Stokes numbers.
Let us notice nevertheless an important difference with respect to the DNS  data. Here the absolute values of the flatness are much smaller, due to the absence of internal intermittency. In the stochastic signal the acceleration of the fluid is non-Gaussian only because of
kinematic effects. In real flows, the acceleration is  more intense and  more fluctuating because of the vortex stretching mechanism and of the turbulent energy cascade.

\section{Conclusions}
\label{sec:conclusions}
In this paper we have analysed a Lagrangian closure  describing fluctuations and correlations
of inertial particle accelerations in turbulent flows in the diluted regime (one-way coupling), i.e. neglecting particle-particle collisions and feedback on the flow. In this way, we have a model that is able to predict some properties of the
acceleration statistics of inertial particles for a large range of values of $\beta$ and $\st$ out of one single measurement based on fluid tracers only. We have compared the predictions of the closure
to DNS of heavy and light inertial particles in turbulence.  To summarize our results, the closure predictions are in overall
good qualitative agreement  with the results of DNS of particles.
The closure neglects inertial preferential sampling, i.e. the tendency of light/heavy particles to be centrifuged in  or out of vortex structures. Hence, the good agreement with the DNS data indicates
 that inertial preferential sampling has in general only a partial effect on inertial particle accelerations.
The main trends are essentially kinematic, a consequence of the form of the equation of motion, as also shown by
the results obtained using a stochastic surrogate for the flow velocity.
A closer inspection shows that there are important differences between the Lagrangian closure scheme and the DNS,
revealing where inertial preferential sampling is important. The effect is larger for light particles
at large Stokes numbers ($\st \sim 10$ in our DNS), and is a consequence of the fact that light bubbles
are drawn into intense vortex tubes. We mention that there is no small-scale fractal clustering for these values of $\st$, i.e. particles are distributed on a three-dimensional set at scales much smaller than the Kolmogorov length.
Finally, non-trivial non-monotonic behaviours of the flatness for both light and heavy particles as a function of $\st$
are predicted by the closure scheme and confirmed by the DNS results, including the fact that light particles are always more intermittent than the fluid tracers and the opposite holds for heavy particles, as shown by the fact that the flatness for the former is always larger than the one of fluid tracers and vice versa for the latter.
The Lagrangian closure scheme must become exact when $\st \to 0$ or $\beta \to 1$, it should therefore be possible to see it
 as a perturbative expansion around Lagrangian tracers and proportional to a small parameter $\epsilon= \st(1-\beta)$,
at least for quantities that depend on Lagrangian correlation functions  decaying on a time scale of the order of the Kolmogorov time.
In this case, one could try to develop an intermediate asymptotic where for small enough time the difference between the two trajectories remains small and then improve the zeroth-order approximation here presented by considering also corrections induced by the velocity gradients around the Lagrangian tracers:
\begin{equation}
 u^i (\ve r_t,t) \sim  u^i (\ve r^{({\rm L})}_t,t) + \partial_i u^j(\ve r^{({\rm L})}_t,t) \delta  r^j_t + \cdots\,,
\end{equation}
where $\delta\ve r=\ve r_t-\ve r_t^{(\rm L)}$.
Work in this direction is in progress.  Finally, we have also investigated the validity of the Lagrangian closure using a stochastic Gaussian surrogate for the advecting fluid velocity field. In such a case, the $\ku$ number is another free parameter that can be tuned to increase/decrease the effects of inertial preferential sampling (effects vanish as $\ku$ approaches zero). We have shown that for large Kubo numbers, corresponding to the long-lived structures in turbulent flows, the closure theory works as well as for the DNS data, even though the data for the statistical model have a much smaller flatness.\\
Let us add some  remarks about the generality and the  limitations of the approach proposed.
First, there are no theoretical difficulties in incorporating  buoyancy, Fax\'{e}n corrections and other forces in the closure scheme as long as the dynamics
 can be described by a point-particle approach. We refrained from presenting here the results because of lack of DNS data to compare with. On the other hand,
 it is known that the equations (\ref{mrfc}) are not valid for all values in the ($\beta,\st$) parameter space. Indeed,
the two requirements that the Reynolds number based on the particle slip velocity is small: $Re_p = |u-v|R/\nu < O(1)$ and that the particle size is smaller
than  the Kolmogorov scale $R/\eta < O(1)$  lead to the condition that $\st < O(1)$ if $\beta > 1$. So the prediction of the model in the limit of large Stokes numbers for light
particles cannot be taken on a quantitative basis.  We stress nevertheless that the most interesting property highlighted by our approach, i.e. the existence of a non-monotonic behaviour for the flatness of the acceleration of light and heavy particles, develops at values of $\st$ where the model equations are still valid. It is difficult to precisely assess the value of Stokes where the approximation breaks down. For instance, recently  it was found \citep{Lohse2015}  that the acceleration variance of light particles with a size up to $R \sim 10\eta $ follow quite closely the point-like approximation (\ref{mrfc}). For even larger particle sizes a wake-driven dynamics  becomes dominant. For such a range of particle parameters no theoretical models for the equations of motion are known.
For instance, recently it was found (Mathai et al. 2015) that the acceleration variance of light particles with a size up to R ∼ 10η follow quite closely the approximate dynamics (2.1).

{\em Acknowledgments.}
This work was supported Vetenskapsr\aa{}det and by the grant {\em Bottlenecks for particle growth in turbulent aerosols} from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048. The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement No 339032.


\section{Introduction}

For $d \in \mathbb{N}$, let $S$ be a finite set of points in $\mathbb{R}^d$.
The set $S$ is in \emph{general position} if, for every $k=1,\dots,d-1$, no $k+2$ points of $S$ lie in an affine $k$-dimensional subspace.
A set $H$ of $k$ points from $S$ is a \emph{$k$-hole} in $S$ if $H$ is in convex position and the interior of the convex hull $\conv(H)$ of $H$ does not contain any point from $S$; see Figure~\ref{fig:preliminaries} for an illustration in the plane.
We say that a subset of $S$ is a \emph{hole} in $S$ if it is a $k$-hole in $S$ for some integer $k$.

\begin{figure}[htb]
\centering
\hbox{}
\hfill
\begin{subfigure}[b]{.3\textwidth}
\centering
\includegraphics[page=1]{figs/preliminaries}
\caption{}
\label{fig:preliminaries_1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.3\textwidth}
\centering
\includegraphics[page=3]{figs/preliminaries}
\caption{}
\label{fig:preliminaries_2}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.3\textwidth}
\centering
\includegraphics[page=2]{figs/preliminaries}
\caption{}
\label{fig:preliminaries_3}
\end{subfigure}
\hfill
\hbox{}

\caption{
(\subref{fig:preliminaries_1})~A $6$-tuple of points in convex position in a planar set $S$ of 10 points.
(\subref{fig:preliminaries_2})~A $6$-hole in $S$.
(\subref{fig:preliminaries_3})~A $6$-island in $S$ whose points are not in convex position.
}
\label{fig:preliminaries}
\end{figure}

Let $h(k)$ be the smallest positive integer $N$ such that every set of $N$ points in general position in the plane contains a $k$-hole.
In the 1970s, Erd\H{o}s~\cite{Erdos1978} asked whether the number $h(k)$ exists for every $k \in \mathbb{N}$.
It was shown in the 1970s and 1980s that $h(4)=5$, $h(5)=10$~\cite{Harborth1978}, and that $h(k)$ does not exist for every $k \geq 7$~\cite{Horton1983}.
That is, while every sufficiently large set contains a $4$-hole and a $5$-hole, Horton constructed arbitrarily large sets with no 7-holes.
His construction was generalized to so-called \emph{Horton sets} by Valtr~\cite{VALTR1992b}.
The existence of 6-holes in every sufficiently large point set remained open until 2007, when Gerken~\cite{Gerken2008} and Nicolas~\cite{Nicolas2007} independently showed that $h(6)$ exists; see also~\cite{Valtr2009}.

These problems were also considered in higher dimensions.
For $d \geq 2$, let $h_d(k)$ be the smallest positive integer $N$ such that every set of $N$ points in general position in~$\mathbb{R}^d$ contains a $k$-hole.
In particular, $h_2(k) = h(k)$ for every $k$.
Valtr~\cite{VALTR1992b} showed that $h_d(k)$ exists for $k \le 2d+1$ but it does not exist for $k > 2^{d-1} (P(d-1)+1)$,  where $P(d-1)$ denotes the product of the first $d-1$ prime numbers.
The latter result was obtained by constructing multidimensional analogues of the Horton sets.

After the existence of $k$-holes was settled, counting the minimum number $H_k(n)$ of $k$-holes in any set of $n$ points in the plane in general position attracted a lot of attention.
It is known, and not difficult to show, that $H_3(n)$ and $H_4(n)$ are in $\Omega(n^2)$.
The currently best known lower bounds on $H_3(n)$ and $H_4(n)$ were proved in~\cite{5holes_Socg2017}.
The best known upper bounds are due to B\'{a}r\'{a}ny and Valtr~\cite{BaranyValtr2004}.
Altogether, these estimates are
\[n^2 + \Omega(n\log^{2/3}n) \le  H_3(n) \le 1.6196 n^2 +o(n^2)\]
and
\[\tfrac{n^2}{2} +\Omega(n\log^{3/4}n)  \le  H_4(n) \le 1.9397 n^2 +o(n^2).\]
For $H_5(n)$ and $H_6(n)$, the best quadratic upper bounds can be found in~\cite{BaranyValtr2004}.
The best lower bounds, however, are only $H_5(n) \geq \Omega(n \log^{4/5}n)$~\cite{5holes_Socg2017}
and $H_6(n) \geq \Omega(n)$~\cite{Valtr2012}.
For more details, we also refer to the second author's dissertation~\cite{Scheucher2019_dissertation}.

The quadratic upper bound on $H_3(n)$ can be also obtained using random point sets.
For $d \in \mathbb{N}$, a \emph{convex body} in $\mathbb{R}^d$ is a compact convex set in $\mathbb{R}^d$ with a nonempty interior.
Let $k$ be a positive integer and let $K \subseteq \mathbb{R}^d$ be a convex body with $d$-dimensional Lebesgue measure $\lambda_d(K)=1$.
We use $EH^K_{d,k}(n)$ to denote the expected number of $k$-holes in sets of $n$ points chosen independently and uniformly at random from $K$.
The quadratic upper bound on $H_3(n)$ then also follows from the following bound of B\'{a}r\'{a}ny and F\"{u}redi~\cite{BaranyFueredi1987} on the expected number of $(d+1)$-holes:
\begin{equation}
\label{boundBaranyFuredi}
EH_{d,d+1}^K(n) \leq (2d)^{2d^2} \cdot \binom{n}{d}
\end{equation}
 for any $d$ and $K$.
In the plane, B\'{a}r\'{a}ny and F\"{u}redi~\cite{BaranyFueredi1987} proved $EH_{2,3}^K(n) \le 2n^2+O(n \log n)$ for every~$K$. This bound was later slightly improved by Valtr~\cite{Valtr1995}, who showed $EH^K_{2,3}(n) \le 4\binom{n}{2}$ for any $K$.
In the other direction, every set of $n$ points in $\mathbb{R}^d$ in general position contains at least $\binom{n-1}{d}$ $(d+1)$-holes~\cite{BaranyFueredi1987,katMe88}.

The expected number $EH_{2,4}^K(n)$ of $4$-holes in random sets of $n$ points in the plane was considered by Fabila-Monroy, Huemer, and Mitsche~\cite{mhm15}, who showed
\begin{equation}
\label{eq-MHM}
EH_{2,4}^K(n) \leq 18\pi D^2 n^2 + o(n^2)
\end{equation}
for any $K$, where $D=D(K)$ is the diameter of $K$.
Since we have $D \geq 2/\sqrt{\pi}$, by the Isodiametric inequality~\cite{evansGariepy15}, the leading constant in~\eqref{eq-MHM} is at least $72$ for any $K$.

In this paper, we study the number of $k$-holes in random point sets in $\mathbb{R}^d$.
In particular, we obtain results that imply quadratic upper bounds on $H_k(n)$ for any fixed $k$ and that both strengthen and generalize the bounds by B\'{a}r\'{a}ny and F\"{u}redi~\cite{BaranyFueredi1987}, Valtr~\cite{Valtr1995}, and Fabila-Monroy, Huemer, and Mitsche~\cite{mhm15}.

\section{Our results}
\label{sec-ourResults}

Throughout the whole paper we only consider point sets in~$\mathbb{R}^d$ that are finite and in general position.

\subsection{Islands and holes in random point sets}

First, we prove a result that gives the estimate $O(n^d)$ on the minimum number of $k$-holes in a set of $n$ points in $\mathbb{R}^d$ for any fixed $d$ and $k$.
In fact, we prove the upper bound $O(n^d)$ even for so-called $k$-islands, which are also frequently studied in discrete geometry.
A set $I$ of $k$ points from a point set $S \subseteq \mathbb{R}^d$ is a \emph{$k$-island} in $S$ if $\conv (I) \cap S = I$; see part~(c) of Figure~\ref{fig:preliminaries}.
Note that $k$-holes in $S$ are exactly those $k$-islands in~$S$ that are in convex position.
A subset of $S$ is an \emph{island} in $S$ if it is a $k$-island in $S$ for some integer $k$.

\begin{theorem}
\label{thm:islands_2d}
Let $d \geq 2$ and $k \geq d+1$ be integers and let $K$ be a convex body in~$\mathbb{R}^d$ with $\lambda_d(K)=1$.
If $S$ is a set of $n \geq k$ points chosen uniformly and independently at random from~$K$, then the expected number of $k$-islands in $S$ is at most
\[2^{d-1}\cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot (k-d) \cdot \frac{n(n-1) \cdots (n-k+2)}{(n-k+1)^{k-d-1}},\]
which is in $O(n^d)$ for any fixed $d$ and $k$.
\end{theorem}

The bound in Theorem~\ref{thm:islands_2d} is tight up to a constant multiplicative factor that depends on $d$ and $k$, as, for any fixed $k \geq d$, every set $S$ of $n$ points in~$\mathbb{R}^d$ in general position contains at least $\Omega(n^d)$ $k$-islands.
To see this, observe that any $d$-tuple $T$ of points from $S$ determines a $k$-island with $k-d$ closest points to the hyperplane spanned by $T$ (ties can be broken by, for example, taking points with lexicographically smallest coordinates), as $S$ is in general position and thus $T$ is a $d$-hole in $S$.
Any such $k$-tuple of points from $S$ contains $\binom{k}{d}$ $d$-tuples of points from $S$ and thus we have at least $\binom{n}{d}/\binom{k}{d} \in \Omega(n^d)$ $k$-islands in $S$.

Thus, by Theorem~\ref{thm:islands_2d}, random point sets in $\mathbb{R}^d$ asymptotically achieve the minimum number of $k$-islands.
This is in contrast with the fact that, unlike Horton sets, they contain arbitrarily large holes.
Quite recently, Balogh, Gonz\'alez-Aguilar, and Salazar~\cite{BaloghGAS2013} showed that the expected number of vertices of the largest hole in a set of $n$ random points chosen independently and uniformly over a convex body in the plane is in $\Theta(\log n/(\log \log n))$.

For $k$-holes, we modify the proof of Theorem~\ref{thm:islands_2d} to obtain a slightly better estimate.

\begin{theorem}
\label{thm:holes_2d}
Let $d \geq 2$ and $k \geq d+1$ be integers and let $K$ be a convex body in~$\mathbb{R}^d$ with $\lambda_d(K)=1$.
If $S$ is a set of $n \geq k$ points chosen uniformly and independently at random from~$K$, then the expected number $EH^K_{d,k}(n)$ of $k$-holes in $S$ is in $O(n^d)$ for any fixed $d$ and $k$.
More precisely,
\[EH^K_{d,k}(n) \leq 2^{d-1}\cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot \frac{n(n-1) \cdots (n-k+2)}{(k-d-1)! \cdot (n-k+1)^{k-d-1}}.\]
\end{theorem}

For $d=2$ and $k=4$, Theorem~\ref{thm:holes_2d} implies $EH^K_{2,4}(n) \leq 128 \cdot n^2 + o(n^2)$ for any $K$, which is a worse estimate than~\eqref{eq-MHM} if the diameter of $K$ is at most $8/(3\sqrt{\pi}) \simeq 1.5$.
However, the proof of Theorem~\ref{thm:holes_2d} can be modified to give $EH^K_{2,4}(n) \leq 12n^2 + o(n^2)$ for any $K$, which is always better than~\eqref{eq-MHM}; see the final remarks in Section~\ref{sec:islands_in_Rd}.
We believe that the leading constant in $EH^K_{2,4}(n)$ can be estimated even more precisely and we hope to discuss this direction in future work.

In the case $k=d+1$, the bound in Theorem~\ref{thm:holes_2d} simplifies to the following estimate on the expected number of $(d+1)$-holes (also called \emph{empty simplices}) in random sets of $n$ points in~$\mathbb{R}^d$.

\begin{corollary}
\label{thm:d_simplices}
Let $d \geq 2$ be an integer and let $K$ be a convex body in $\mathbb{R}^d$ with $\lambda_d(K)=1$.
If $S$ is a set of $n$ points chosen uniformly and independently at random from~$K$, then the expected number of $(d+1)$-holes in $S$
satisfies
\[EH^K_{d,d+1}(n) \leq 2^{d-1} \cdot d! \cdot \binom{n}{d}.\]
\end{corollary}

Corollary~\ref{thm:d_simplices} is stronger than
the bound~\eqref{boundBaranyFuredi} by B\'{a}r\'{a}ny and F\"{u}\-redi~\cite{BaranyFueredi1987} and,
in the planar case, coincides with the bound $EH^K_{2,3}(n) \le 4\binom{n}{2}$ by Valtr~\cite{Valtr1995}.
Very recently, Reitzner and Temesvari \cite{ReitznerTemesvari2019}
proved an upper bound on $EH^K_{d,d+1}(n)$ that is asymptotically tight if $d=2$ or if $d \ge 3$ and $K$ is an ellipsoid.
In the planar case, their result shows that the bound $4\binom{n}{2}$ on $EH^K_{2,3}(n)$ is best possible, up to a smaller order error term.

We also consider islands of all possible sizes and show that their expected number is in $2^{\Theta\left(n^{(d-1)/(d+1)}\right)}$.

\begin{theorem}
\label{thm:exponential}
Let $d \geq 2$ be an integer and let $K$ be a convex body in $\mathbb{R}^d$ with $\lambda_d(K)=1$.
Then there are constants $C_1=C_1(d)$, $C_2=C_2(d)$, and $n_0=n_0(d)$ such that for every set $S$ of $n \geq n_0$ points chosen uniformly and  independently at random from $K$ the expected number $E^K_d$ of islands in $S$ satisfies
\[2^{C_1 \cdot n^{(d-1)/(d+1)}} \leq E^K_d \leq 2^{C_2 \cdot n^{(d-1)/(d+1)}}.\]
\end{theorem}

Since each island in $S$ has at most $n$ points, there is a $k \in \{1,\dots,n\}$ such that the expected number of $k$-islands in $S$ is at least $(1/n)$-fraction of the expected number of all islands, which is still in $2^{\Omega(n^{(d-1)/(d+1)})}$.
This shows that the expected number of $k$-islands can become asymptotically much larger than $O(n^d)$ if $k$ is not fixed.

\subsection{Islands and holes in $d$-Horton sets}

To our knowledge, Theorem~\ref{thm:islands_2d} is the first nontrivial upper bound on the minimum number of $k$-islands a point set in $\mathbb{R}^d$ with $d>2$ can have.
For $d=2$, Fabila-Monroy and Huemer~\cite{FabilaMonroyHuemer2012} showed that, for every fixed $k \in\mathbb{N}$, the Horton sets with $n$ points contain only $O(n^2)$ $k$-islands.
For $d>2$, Valtr~\cite{VALTR1992b} introduced a $d$-dimensional analogue of Horton sets.
Perhaps surprisingly, these sets contain asymptotically more than $O(n^d)$ $k$-islands for $k \geq d+1$.
For each $k$ with $d+1 \leq k \leq 3 \cdot 2^{d-1}$, they even contain asymptotically more than $O(n^d)$ $k$-holes.

\begin{theorem}
\label{prop-Horton}
Let $d \geq 2$ and $k$ be fixed positive integers.
Then every $d$-dimensional Horton set $H$ with $n$ points contains at least $\Omega(n^{\min\{2^{d-1},k\}})$ $k$-islands in $H$.
If $k \leq 3 \cdot 2^{d-1}$, then $H$ even contains at least $\Omega(n^{\min\{2^{d-1},k\}})$ $k$-holes in $H$.
\end{theorem}

\section{Proofs of Theorem~\ref{thm:islands_2d} and Theorem~\ref{thm:holes_2d}}
\label{sec:islands_in_Rd}

Let $d$ and $k$ be positive integers and let $K$ be a convex body in~$\mathbb{R}^d$ with $\lambda_d(K)=1$.
Let $S$ be a set of $n$ points chosen uniformly and independently at random from~$K$.
Note that $S$ is in general position with probability $1$.
We assume $k \geq d+1$, as otherwise the number of $k$-islands in $S$ is trivially $\binom{n}{k}$ in every set of $n$ points in $\mathbb{R}^d$ in general position.
We also assume $d \geq 2$ and $n \geq k$, as otherwise the number of $k$-islands is trivially $n-k+1$ and $0$, respectively, in every set of $n$ points in $\mathbb{R}^d$.

First, we prove Theorem~\ref{thm:islands_2d} by showing that the expected number of $k$-islands in $S$ is at most
\[2^{d-1}\cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot (k-d) \cdot \frac{n(n-1) \cdots (n-k+2)}{(n-k+1)^{k-d-1}},\]
which is in $O(n^d)$ for any fixed $d$ and $k$.
At the end of this section, we improve the bound for $k$-holes, which will prove Theorem~\ref{thm:holes_2d}.

Let $Q$ be a set of $k$ points from $S$.
We first introduce a suitable unique ordering $q_1,\dots,q_k$ of points from~ $Q$.
First, we take a set $D$ of $d+1$ points from $Q$ that determine a simplex $\Delta$ with largest volume among all $(d+1)$-tuples of points from $Q$.
Let $q_1q_2$ be the longest edge of $\Delta$ with $q_1$ lexicographically smaller than $q_2$ and let $a$ be the number of points from $Q$ inside $\Delta$.
For every $i=2,\dots,d$, let $q_{i+1}$ be the furthest point from $D\setminus\{q_1,\dots,q_i\}$ to ${\rm aff}(q_1,\dots,q_i)$.
Next, we let $q_{d+2},\dots,q_{d+a+1}$ be the $a$ points of $Q$ inside $\Delta$ ordered lexicographically.
The remaining $k-d-a-1$ points $q_{d+a+2},\dots,q_k$ from $Q$ lie outside of $\Delta$ and we order them so that, for every $i=1,\dots,k-a-d-1$, the point $q_{d+a+i+1}$ is closest to ${\rm conv}(\{q_1,\dots,\allowbreak q_{d+a+i}\})$ among the points $q_{d+a+i+1},\dots,q_k$.
In case of a tie in any of the conditions, we choose the point with lexicographically smallest coordinates.
Note, however, that a tie occurs with probability $0$.

Clearly, there is a unique such ordering $q_1,\dots,q_k$ of $Q$.
We call this ordering the \emph{canonical $(k,a)$-ordering} of $Q$.
To reformulate, an ordering $q_1,\dots,q_k$ of $Q$ is the canonical $(k,a)$-ordering of~$Q$ if and only if the following five conditions are satisfied:
\begin{enumL}{L}
\item\label{item-canonical0} The $d$-dimensional simplex $\Delta$,  with vertices $q_1,\dots,q_{d+1}$ has the largest $d$-dimen\-sional Lebesgue measure among all $d$-dimensional simplices spanned by points from~$Q$.

\item\label{item-canonical1}
For every $i=1,\dots,d-1$, the point $q_{i+1}$ has the largest distance among all points from $\{q_{i+1},\dots,q_d\}$ to the $(i-1)$-dimensional affine subspace ${\rm aff}(q_1,\dots,q_i)$ spanned by $q_1,\dots,q_i$.
Moreover, $q_1$ is lexicographically smaller than $q_2$.

\item\label{item-canonical4} For every $i=1,\dots,d-1$, the distance between $q_{i+1}$ and ${\rm aff}(q_1,\dots,q_{i})$ is at least as large as the distance between $q_{d+1}$ and ${\rm aff}(q_1,\dots,q_i)$.
Also, the distance between $q_1$ and $q_2$ is at least as large as the distance between $q_{d+1}$ and any $q_i$ with $i \in \{1,\dots,d\}$.

\item\label{item-canonical2} The points $q_{d+2},\dots,q_{d+a+1}$ lie inside $\Delta$ and are ordered lexicographically.

\item\label{item-canonical3} The points $q_{d+a+2},\dots,q_k$ lie outside of $\Delta$.
For every $i=1,\dots,k-a-d-1$, the point $q_{d+a+i+1}$ is closest to ${\rm conv}(\{q_1,\dots,\allowbreak q_{d+a+i}\})$ among the points $q_{d+a+i+1},\dots,q_k$.
\end{enumL}

Figure~\ref{fig:canlab} gives an illustration in~$\mathbb{R}^2$.
We note that the conditions~\ref{item-canonical1} and~\ref{item-canonical4} can be merged together.
However, later in the proof, we use the fact that the probability that the points from $Q$ satisfy the condition~\ref{item-canonical1} equals $1/d!$, so we stated the two conditions separately.

\begin{figure}[htb]
  \centering
    \includegraphics{figs/canlab}
  \caption{An illustration of the canonical $(k,a)$-ordering of a planar point set~$Q$.
  Here we have $k=12$ points and $a=4$ of the points
  lie inside the largest area triangle $\triangle$ with vertices $q_1,q_2,q_3$.}
  \label{fig:canlab}
\end{figure}

Before going into details, we first give a high-level overview of the proof of Theorem~\ref{thm:islands_2d}.
First, we prove an $O(1/n^{a+1})$ bound on the probability that
$\triangle$ contains precisely the points $p_{d+2},\ldots,p_{d+1+a}$ from~$S$ (Lemma~\ref{lem-probabilityInside}),
which means that the points $p_1,\ldots,p_{d+1+a}$ determine an island in $S$.
Next, for $i=d+2+a,\ldots,k$,
we show that,
conditioned on the fact that
the $(i-1)$-tuple $(p_1,\ldots,p_{i-1})$ determines an island in $S$
in the canonical $(k,a)$-ordering,
the $i$-tuple $(p_1,\ldots,p_{i})$ determines an island in $S$
in the canonical $(k,a)$-ordering
with probability $O(1/n)$ (Lemma~\ref{lem-probabilityOutside}).
Then it immediately follows that the probability
that $I$ determines a $k$-island in $S$ with the desired properties is at most $
O \left( 1/n^{a+1} \cdot (1/n)^{k-(d+1+a)} \right) = O(n^{d-k})$.
Since there are $n \cdot (n-1)\cdots(n-k+1)=O(n^k)$
possibilities to select such an ordered subset $I$
and each $k$-island in $S$ is counted at most $k!$ times,
we obtain the desired bound $O \left( n^k \cdot n^{d-k} \cdot k!  \right) = O(n^d)$
on the expected number of $k$-islands in~$S$.

We now proceed with the proof.
Let $p_1,\dots,p_k$ be points from $S$ in the order in which they are drawn from $K$.
We use $\Delta$ to denote the $d$-dimensional simplex with vertices $p_1,\dots,p_{d+1}$.
We eventually show that the probability that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering of a $k$-island in $S$ for some $a$ is at most $O(1/n^{k-d})$.
First, however, we need to state some notation and prove some auxiliary results.

Consider the points $p_1,\dots,p_d$.
Without loss of generality, we can assume that, for each $i=1,\dots,d$, the point $p_i$ has the last $d-i+1$ coordinates equal to zero.
Otherwise we apply a suitable isometry to~$S$.
Then, for every $i=1,\dots,d$, the distance between $p_{i+1}$ and the $(i-1)$-dimensional affine subspace spanned by $p_1,\dots,p_i$ is equal to the absolute value of the $i$th coordinate of $p_{i+1}$.
Moreover, after applying a suitable rotation, we can also assume that the first coordinate of each of the points $p_1,\dots,p_d$ is nonnegative.

Let $\Delta_0$ be the $(d-1)$-dimensional simplex with vertices $p_1,\dots,p_d$ and let $H$ be the hyperplane containing $\Delta_0$.
Note that, according to our assumptions about $p_1,\dots,p_d$, we have $H = \{(x_1,\dots,x_d) \in \mathbb{R}^d \colon x_d=0\}$.
Let $B$ be the set of points $(x_1,\dots,x_d) \in \mathbb{R}^d$ that satisfy the following three conditions:
\begin{enumerate}[label=(\roman*)]
\item $x_1 \geq 0$,
\item $|x_i|$ is at most as large as the absolute value of the $i$th coordinate of $p_{i+1}$ for every $i \in \{1,\dots,d-1\}$, and
\item $|x_d| \leq d/\lambda_{d-1}(\Delta_0)$.
\end{enumerate}
See Figures~\ref{fig:simplices_2d} and~\ref{fig:simplices_3d} for illustrations in $\mathbb{R}^2$ and $\mathbb{R}^3$, respectively.
Observe that $B$ is a $d$-dimensional axis-parallel box.
For $h \in \mathbb{R}$, we use $I_h$ to denote the intersection of $B$ with the hyperplane $x_d=h$.

\begin{figure}[htb]
\centering
\hbox{}
\hfill
\begin{subfigure}[b]{.35\textwidth}
\centering
    \includegraphics{figs/simplices_2d.pdf}
\caption{}
\label{fig:simplices_2d}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.5\textwidth}
\centering
    \includegraphics{figs/simplices_3d.pdf}
\caption{}
\label{fig:simplices_3d}
\end{subfigure}
\hfill
\hbox{}

\caption{
An illustration of the proof of Theorem~\ref{thm:islands_2d} in~(\subref{fig:simplices_2d})~$\mathbb{R}^2$ and (\subref{fig:simplices_3d})~$\mathbb{R}^3$.
}
\end{figure}

Having fixed $p_1,\dots,p_d$, we now try to restrict possible locations of the points $p_{d+1},\dots,p_k$, one by one, so that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering of a $k$-island in $S$ for some $a$.
First, we observe that the position of the point $p_{d+1}$ is restricted to $B$.

\begin{lemma}
\label{lem-containment}
If $p_1,\dots,p_{d+1}$ satisfy condition~\ref{item-canonical4}, then $p_{d+1}$ lies in the box~$B$.
\end{lemma}
\begin{proof}
Let $p_{d+1}=(x_1,\dots,x_d)$.
According to our choice of points $p_1,\dots,p_d$ and from the assumption that $p_1,\dots,p_d$ satisfy~\ref{item-canonical4}, we get $x_1 \geq 0$ and also that $|x_i|$ is at most as large as the absolute value of the $i$th coordinate of $p_{i+1}$ for every $i \in \{1,\dots,d-1\}$.

It remains to show that $|x_d| \leq d/\lambda_{d-1}(\Delta_0)$.
The simplex $\Delta$ spanned by $p_1,\dots,p_{d+1}$ is contained in the convex body $K$, as $p_1,\dots,p_{d+1} \in K$ and $K$ is convex.
Thus $\lambda_d(\Delta) \leq \lambda_d(K) = 1$.
On the other hand, the volume $\lambda_d(\Delta)$ equals $\lambda_{d-1}(\Delta_0) \cdot h/d$, where $h$ is the distance between $p_{d+1}$ and the hyperplane $H$ containing $\Delta_0$.
According to our assumptions about $p_1,\dots,p_d$, the distance $h$ equals $|x_d|$.
Since $\lambda_d(\Delta) \leq 1$, it follows that $|x_d| = h \leq d/\lambda_{d-1}(\Delta_0)$ and thus $p_{d+1} \in B$.
\end{proof}

The following auxiliary lemma gives an identity that is needed later.
We omit the proof, which can be found, for example, in~\cite[Section~1]{aar99}.

\begin{lemma}[\cite{aar99}]
\label{lem-integral}
For all nonnegative integers $a$ and $b$, we have
\[\int_0^1 x^a(1-x)^b \;{\rm d}x = \frac{a! \; b!}{(a+b+1)!}\,.\]
\end{lemma}

We will also use the following result, called the \emph{Asymptotic Upper Bound Theorem}~\cite{mat02}, that estimates the maximum number of facets in a polytope.

\begin{theorem}[Asymptotic Upper Bound Theorem~\cite{mat02}]
\label{thm-upperBoundThm}
For every integer $d \geq 2$, a $d$-dimensional convex polytope with $N$ vertices has at most $2\binom{N}{\lfloor d/2\rfloor}$ facets.
\end{theorem}

Let $a$ be an integer satisfying $0 \leq a \leq k-d-1$ and let $E_a$ be the event that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering such that $\{p_1,\dots,p_{d+a+1}\}$ is an island in $S$.
To estimate the probability that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering of a $k$-island in $S$, we first find an upper bound on the conditional probability of~$E_a$, conditioned on the event $L_2$ that $p_1,\dots,p_d$ satisfy~\ref{item-canonical1}.

\begin{lemma}
\label{lem-probabilityInside}
For every $a \in \{0,\dots,k-d-1\}$, the probability $\Pr[E_a \mid L_2]$ is at most
\[\frac{2^{d-1}\cdot d!}{(k-a-d-1)!\cdot(n-k+1)^{a+1}} .\]
\end{lemma}
\begin{proof}

It follows from Lemma~\ref{lem-containment} that, in order to satisfy~\ref{item-canonical4}, the point $p_{d+1}$ must lie in the box~$B$.
In particular, $p_{d+1}$ is contained in $I_h \cap K$ for some real number $h \in [-d/\lambda_{d-1}(\Delta_0),d/\lambda_{d-1}(\Delta_0)]$.
If $p_{d+1} \in I_h$, then the simplex $\Delta = \conv(\{p_1,\dots,p_{d+1}\})$ has volume $\lambda_d(\Delta) = \lambda_{d-1}(\Delta_0)\cdot|h|/d$ and the $a$ points $p_{d+2},\dots,p_{d+a+1}$ satisfy~\ref{item-canonical2} with probability
\[\frac{1}{a!}\cdot \left(\lambda_d(\Delta)\right)^a = \frac{1}{a!}\cdot \left(\frac{\lambda_{d-1}(\Delta_0) \cdot |h|}{d}\right)^a,\] as they all lie in $\Delta \subseteq K$ in the unique order.

In order to satisfy the condition~\ref{item-canonical3},
the $k-a-d-1$ points $p_{d+a+i+1}$, for $i \in \{1,\dots,k-a-d-1\}$, must have increasing distance to ${\rm conv}(\{p_1,\dots,\allowbreak p_{d+a+i}\})$ as the index $i$ increases, which happens with probability at most $\frac{1}{(k-a-d-1)!}$.
Since $\{p_1,\dots,p_{d+a+1}\}$ must be an island in $S$, the $n-d-a-1$ points from $S \setminus \{p_1,\dots,p_{d+a+1}\}$ must lie outside $\Delta$.
If $p_{d+1} \in I_h$, then this happens with probability
\[(\lambda_d(K \setminus \Delta))^{n-d-a-1} = (\lambda_d(K)-\lambda_d(\Delta))^{n-d-a-1} = \left(1 - \frac{\lambda_{d-1}(\Delta_0) \cdot |h|}{d}\right)^{n-d-a-1},\]
as they all lie in $K \setminus \Delta$ and we have $\Delta \subseteq K$ and $\lambda_d(K)=1$.

Altogether, we get that $\Pr[E_a \mid L_2]$ is at most
\[\int\displaylimits_{-d/\lambda_{d-1}(\Delta_0)}^{d/\lambda_{d-1}(\Delta_0)} \frac{\lambda_{d-1}(I_h \cap K)}{a!\cdot (k-a-d-1)!} \cdot \left(\frac{\lambda_{d-1}(\Delta_0) \cdot |h|}{d}\right)^a \cdot \left(1 - \frac{\lambda_{d-1}(\Delta_0) \cdot |h|}{d}\right)^{n-d-a-1} {\rm d} h.
 \]
Since we have $\lambda_{d-1}(I_0) = \lambda_{d-1}(I_h)$ for every $h \in [-d/\lambda_{d-1}(\Delta_0),d/\lambda_{d-1}(\Delta_0)]$, we obtain $\lambda_{d-1}(I_h \cap K) \leq \lambda_{d-1}(I_0)$ and thus $\Pr[E_a \mid L_2]$ is at most
\[\frac{2 \cdot \lambda_{d-1}(I_0)}{a!\cdot (k-a-d-1)!   } \cdot \int\displaylimits_0^{d/\lambda_{d-1}(\Delta_0)} \left(\frac{\lambda_{d-1}(\Delta_0) \cdot h}{d}\right)^a \cdot \left(1 - \frac{\lambda_{d-1}(\Delta_0) \cdot h}{d}\right)^{n-d-a-1} {\rm d} h.\]

By substituting $t=\frac{\lambda_{d-1}(\Delta_0) \cdot h}{d}$, we obtain
\[\Pr[E_a \mid L_2] \leq \frac{2d\cdot \lambda_{d-1}(I_0)}{a!\cdot (k-a-d-1)!\cdot\lambda_{d-1}(\Delta_0)} \cdot  \int_0^1 t^a (1-t)^{n-d-a-1} {\rm d}t.\]
By Lemma~\ref{lem-integral}, the right side in the above inequality equals
\begin{align*}
\frac{2d\cdot \lambda_{d-1}(I_0)}{a!\cdot (k-a-d-1)! \cdot \lambda_{d-1}(\Delta_0)} &\cdot \frac{a! \cdot (n-d-a-1)!}{(n-d)!} \\
&= \frac{2d\cdot\lambda_{d-1}(I_0)}{(k-a-d-1)! \cdot\lambda_{d-1}(\Delta_0)} \cdot \frac{(n-d-a-1)!}{(n-d)!}.
\end{align*}

For every $i=1,\dots,d-1$, let $h_i$ be the distance between the point $p_{i+1}$ and the $(i-1)$-dimensional affine subspace spanned by $p_1,\dots,p_i$.
Since the volume of the box $I_0$ satisfies
\[\lambda_{d-1}(I_0) = h_1(2h_2)\cdots(2h_{d-1}) = 2^{d-2} \cdot h_1\cdots h_{d-1}\]
and the volume of the $(d-1)$-dimensional simplex $\Delta_0$ is
\[\lambda_{d-1}(\Delta_0) = \frac{h_1}{1} \cdot \frac{h_2}{2} \cdot \, \cdots \, \cdot \frac{h_{d-1}}{d-1} = \frac{h_1 \cdots h_{d-1}}{(d-1)!},\]
we obtain $\lambda_{d-1}(I_0) / \lambda_{d-1}(\Delta_0) = 2^{d-2}\cdot (d-1)!$.
Thus
\begin{align*}
\Pr[E_a \mid L_2] &\leq \frac{  2^{d-1} \cdot d!  }{(k-a-d-1)!}    \cdot \frac{(n-d-a-1)!}{(n-d)!} \\
&= \frac{2^{d-1}\cdot d!}{(k-a-d-1)! \cdot (n-d)\cdots(n-d-a)}\\
&\leq \frac{2^{d-1}\cdot d!}{(k-a-d-1)! \cdot    (n-k+1)^{a+1} },
\end{align*}
where the last inequality follows from $a \leq k-d-1$.
\end{proof}

For every $i \in \{d+a+1,\dots,k\}$, let $E_{a,i}$ be the event that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering such that  $\{p_1,\dots,p_i\}$ is an island in $S$.
Note that in the event $E_{a,i}$ the condition~\ref{item-canonical3} implies that $\{p_1,\dots,p_j\}$ is an island in $S$ for every $j \in \{d+a+1,\dots,i\}$.
Thus we have
\[L_2 \supseteq E_a = E_{a,d+a+1} \supseteq E_{a,d+a+2} \supseteq \cdots \supseteq E_{a,k}.\]
Moreover, the event $E_{a,k}$ says that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering of a $k$-island in~$S$.

For $i \in \{d+a+2,\dots,k\}$, we now estimate the conditional probability of $E_{a,i}$, conditioned on $E_{a,i-1}$.

\begin{lemma}
\label{lem-probabilityOutside}
For every $i \in \{d+a+2,\dots,k\}$, we have
\[\Pr[E_{a,i} \mid E_{a,i-1}] \leq \frac{2d^{2d-1} \cdot  \binom{k}{\lfloor d/2 \rfloor}}{n-i+1}.\]
\end{lemma}
\begin{proof}
Let $i \in \{d+a+2,\dots,k\}$ and assume that the event $E_{a,i-1}$ holds.
That is, $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering such that $\{p_1,\dots,p_{i-1}\}$ is an $(i-1)$-island in $S$.

First, we assume that $\Delta$ is a regular simplex with height $\eta>0$.
At the end of the proof we show that the case when $\Delta$ is an arbitrary simplex follows by applying a suitable affine transformation.

For every $j \in \{1,\dots,d+1\}$, let $F_j$ be the facet $\conv(\{p_1,\dots,\allowbreak p_{d+1}\}\setminus\{p_j\})$ of $\Delta$ and let $H_j$ be the hyperplane parallel to $F_j$ that contains $p_j$.
We use $H^+_j$ to denote the halfspace determined by $H_j$ such that $\Delta \subseteq H_j^+$.
We set $\Delta^* = \cap_{j=1}^{d+1} H^+_j$; see Figures~\ref{fig:Fischer2d} and \ref{fig:Fischer3d} for illustrations in $\mathbb{R}^2$ and~$\mathbb{R}^3$, respectively.
Note that $\Delta^*$ is a $d$-dimensional simplex containing~$\Delta$.
Also, notice that if $x \notin \Delta^*$, then $x \notin H^+_j$ for some $j$ and the distance between $x$ and the hyperplane containing $F_j$ is larger than $\eta$.

\begin{figure}[htb]
\centering
\hbox{}
\hfill
\begin{subfigure}[b]{.45\textwidth}
\centering
    \includegraphics{figs/Fischer2d}
\caption{}
\label{fig:Fischer2d}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.5\textwidth}
\centering
    \includegraphics[width=0.95\textwidth]{figs/Fischer3d}
\caption{}
\label{fig:Fischer3d}
\end{subfigure}
\hfill
\hbox{}

\caption{
An illustration of
(\subref{fig:Fischer2d})~the simplex $\Delta^*$ in $\mathbb{R}^2$ and
(\subref{fig:Fischer3d})~in $\mathbb{R}^3$.
}
\end{figure}

We show that the fact that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering implies that every point from $\{p_1,\dots,p_k\}$ is contained in $\Delta^*$.
Suppose for contradiction that some point $p \in \{p_1,\dots,p_k\}$ does not lie inside $\Delta^*$.
Then there is a facet $F_j$ of $\Delta$ such that the distance $\eta'$ between $p$ and the hyperplane containing $F_j$ is larger than $\eta$.
Then, however, the simplex $\Delta'$ spanned by vertices of $F_j$ and by $p$ has volume larger than $\Delta$, because
\[\lambda_d(\Delta') = \frac{1}{d} \cdot \lambda_{d-1}(F_j) \cdot \eta' > \frac{1}{d} \cdot \lambda_{d-1}(F_j) \cdot \eta = \lambda_d(\Delta).\]
This contradicts the fact that $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering, as, according to~\ref{item-canonical0}, $\Delta$ has the largest $d$-dimensional Lebesgue measure among all $d$-dimensional simplices spanned by points from $\{p_1,\dots,p_k\}$.

Let $\sigma$ be the barycenter of $\Delta$.
For every point $p \in \Delta^* \setminus \Delta$, the line segment $\sigma p$ intersects at least one facet of $\Delta$.
For every $j \in \{1,\dots,d+1\}$, we use $R_j$ to denote the set of points $p \in \Delta^* \setminus \Delta$ for which the line segment $\sigma p$ intersects the facet $F_j$ of $\Delta$.
Observe that each set $R_j$ is convex and the sets $R_1,\dots,R_{d+1}$ partition $\Delta^* \setminus \Delta$ (up to their intersection of $d$-dimensional Lebesgue measure $0$); see Figure~\ref{fig:Fischer2d_2} for an illustration in the plane.

\begin{figure}[htb]
  \centering
    \includegraphics[page=3]{figs/Fischer2d}
  \caption{An illustration of the proof of Lemma~\ref{lem-probabilityOutside}.
  In order for $\{p_1,\dots,p_i\}$ to be an $i$-island in~$S$, the light gray part cannot contain points from $S$.
  We estimate the probability of this event from above by the probability that the dark gray simplex $\conv(\varphi \cup \{p_i\})$ contains no point of $S$.
  Note that the parameters $\eta$ and $\tau$ coincide for $d=2$, as then $\tau = \frac{d^2-1}{d+1}\eta = \eta$.}
  \label{fig:Fischer2d_2}
\end{figure}

Consider the point $p_i$.
Since $p_1,\dots,p_k$ is the canonical $(k,a)$-ordering, the condition~\ref{item-canonical3} implies that $p_i$ lies outside of the polytope $\conv(\{p_1,\dots,p_{i-1}\})$.
To bound the probability $\Pr[E_{a,i} \mid E_{a,i-1}]$, we need to estimate the probability that $\conv(\{p_1,\dots,p_i\}) \setminus \conv(\{p_1,\dots,p_{i-1}\})$ does not contain any point from $S\setminus \{p_1,\dots,p_i\}$, conditioned on $E_{a,i-1}$.
We know that $p_i$ lies in $\Delta^*\setminus \Delta$ and that $p_i \in R_j$ for some $j \in \{1,\dots,d+1\}$.

Since $p_i \notin \conv(\{p_1,\dots,p_{i-1}\})$, there is a facet $\varphi$ of the polytope $\conv(\{p_1,\dots,p_{i-1}\})$ contained in the closure of $R_j$ such that $\sigma p_i$ intersects $\varphi$.
Since $S$ is in general position with probability $1$, we can assume that $\varphi$ is a $(d-1)$-dimensional simplex.
The point $p_i$ is contained in the convex set $C_\varphi$ that contains all points $c \in \mathbb{R}^d$ such that the line segment $\sigma c$ intersects $\varphi$.
We use $H(0)$ to denote the hyperplane containing $\varphi$.
For a positive $r \in \mathbb{R}$, let $H(r)$ be the hyperplane parallel to $H(0)$ at distance $r$ from $H(0)$ such that $H(r)$ is contained in the halfspace determined by $H(0)$ that does not contain $\conv(\{p_1,\dots,p_{i-1}\})$.
Then we have $p_i \in H(h)$ for some positive $h \in \mathbb{R}$.

Since $p_i \in K$ and $\varphi \subseteq K$, the convexity of $K$ implies that the simplex $\conv(\varphi \cup \{p_i\})$ has volume $\lambda_d(\conv(\varphi \cup \{p_i\})) \leq \lambda_d(K) = 1$.
Since $\lambda_d(\conv(\varphi \cup \{p_i\})) = \lambda_{d-1}(\varphi) \cdot h/d$, we obtain $h \leq  d/\lambda_{d-1}(\varphi)$.

The point $p_i$ lies in the $(d-1)$-dimensional simplex $C_\varphi \cap H(h)$, which is a scaled copy of~$\varphi$.
We show that
\begin{equation}
\label{eq-simplex}
\lambda_{d-1}(C_\varphi \cap H(h)) \leq d^{2d-2}\cdot\lambda_{d-1}(\varphi).
\end{equation}
Let $h_\varphi$ be the distance between $H(0)$ and $\sigma$ and, for every $j \in \{1,\dots,d+1\}$, let $\overline{H}_j$ be the hyperplane parallel to $F_j$ containing the vertex $H_1 \cap \cdots \cap H_{j-1} \cap H_{j+1} \cap \cdots \cap H_{d+1}$ of~$\Delta^*$.
We denote by $\overline{H}^+_j$ the halfspace determined by $\overline{H_j}$ containing $\Delta^*$.
Since $\Delta$ lies on the same side of $H(0)$ as $\sigma$, we see that $h_{\varphi}$ is at least as large as the distance between $\sigma$ and $F_j$, which is $\eta/(d+1)$.
Since $p_i$ lies in $\Delta^* \subseteq \overline{H}^+_j$, we see that $h$ is at most as large as the distance $\tau$ between $\overline{H}_j$ and the hyperplane containing the facet $F_j$ of $\Delta$.
Note that $\tau + \eta/(d+1)$ is the distance of the barycenter of $\Delta^*$ and a vertex of $\Delta^*$ and $d\eta/(d+1)$ is the distance of the barycenter of $\Delta^*$ and a facet of $\Delta^*$.
Thus we get $\tau = \frac{d^2\eta}{d+1} - \frac{\eta}{d+1} = \frac{d^2-1}{d+1}\eta$ from the fact that the distance between the barycenter of a $d$-dimensional simplex and any of its vertices is $d$-times as large as the distance between the barycenter and a facet.
Consequently, $h \leq \frac{d^2-1}{d+1}\eta$ and $\frac{\eta}{d+1} \leq h_\varphi$, which implies $h \leq (d^2-1)h_\varphi$.
Thus $C_\varphi \cap H(h)$ is a scaled copy of~$\varphi$ by a factor of size at most~$d^2$.
This gives $\lambda_{d-1}(C_\varphi \cap H(h)) \leq d^{2d-2}\cdot\lambda_{d-1}(\varphi)$.

Since the simplex $\conv(\varphi \cup \{p_i\})$ is a subset of the closure of $\conv(\{p_1,\dots,p_i\}) \setminus \conv(\{p_1,\dots,p_{i-1}\})$, the probability $\Pr[E_{a,i} \mid E_{a,i-1}]$ can be bounded from above by the conditional probability of the event $A_{i,\varphi}$ that $p_i \in C_\varphi \cap K$ and that no point from $S\setminus \{p_1,\dots,p_i\}$ lies in $\conv(\varphi \cup \{p_i\})$, conditioned on $E_{a,i-1}$.
All points from $S\setminus \{p_1,\dots,p_i\}$ lie outside of $\conv(\varphi \cup \{p_i\})$ with probability
\[\left(   1- \frac{\lambda_d(\conv(\varphi \cup \{p_i\}))}{\lambda_d(K \setminus \conv(\{p_1,\ldots,p_{i-1}\}))}    \right)^{n-i} .\]
Since $\lambda_d(K \setminus \conv(\{p_1,\ldots,p_{i-1}\}))\leq \lambda_d(K) = 1$, this is bounded from above by
\[(1-\lambda_d(\conv(\varphi \cup \{p_i\})))^{n-i} = \left(1-\frac{\lambda_{d-1}(\varphi) \cdot h}{d}\right)^{n-i}.\]
Since the sets $C_\varphi$ partition $K \setminus \conv(\{p_1,\dots,p_{i-1}\})$ (up to intersections of $d$-dimensional Lebesgue measure $0$) and since $h \leq  d/\lambda_{d-1}(\varphi)$, we have, by the law of total probability,
\begin{align*}
\Pr[E_{a,i} \mid E_{a,i-1}] &\leq \sum_{\varphi}\Pr[A_{i,\varphi} \mid E_{a,i-1}]\\
&\leq \sum_\varphi \int\displaylimits_0^{d/\lambda_{d-1}(\varphi)} \lambda_{d-1}(C_\varphi \cap H(h)) \cdot \left(1-\frac{\lambda_{d-1}(\varphi) \cdot h}{d}\right)^{n-i} \; {\rm d}h.
\end{align*}
The sums in the above expression are taken over all facets $\varphi$ of the convex polytope $\conv(\{p_1,\dots,\allowbreak p_{i-1}\})$.
Using~\eqref{eq-simplex}, we can estimate $\Pr[E_{a,i} \mid E_{a,i-1}]$ from above by
\[d^{2d-2} \cdot \sum_\varphi \lambda_{d-1}(\varphi) \cdot \int\displaylimits_0^{d/\lambda_{d-1}(\varphi)}  \left(1-\frac{\lambda_{d-1}(\varphi) \cdot h}{d}\right)^{n-i} \; {\rm d}h.\]
By substituting $t= \frac{\lambda_{d-1}(\varphi) \cdot h}{d}$, we can rewrite this expression as
\[d^{2d-2} \cdot \sum_\varphi \frac{d \cdot \lambda_{d-1}(\varphi)}{\lambda_{d-1}(\varphi)} \cdot \int_0^1  (1-t)^{n-i} \; {\rm d}t = d^{2d-1} \cdot \sum_\varphi \int_0^1 1 \cdot (1-t)^{n-i} \; {\rm d}t.\]
By Lemma~\ref{lem-integral}, this equals
\[d^{2d-1} \cdot \sum_\varphi \frac{0! \cdot (n-i)!}{(n-i+1)!} = \frac{d^{2d-1}}{n-i+1} \sum_\varphi 1.\]
Since $\conv(\{p_1,\dots,\allowbreak p_{i-1}\})$ is a convex polytope in $\mathbb{R}^d$ with at most $i-1 \leq k$ vertices,
Theorem~\ref{thm-upperBoundThm} implies that the number of facets $\varphi$ of $\conv(\{p_1,\dots,\allowbreak p_{i-1}\})$ is at most $2 \binom{k}{\lfloor d/2 \rfloor}$.
Altogether, we have derived the desired bound
\[\Pr[E_{a,i} \mid E_{a,i-1}] \leq \frac{2d^{2d-1} \cdot  \binom{k}{\lfloor d/2 \rfloor}}{n-i+1}\]
in the case when $\Delta$ is a regular simplex.

If $\Delta$ is not regular, we first apply a volume-preserving affine transformation $F$ that maps $\Delta$ to a regular simplex $F(\Delta)$.
The simplex $F(\Delta)$ is then contained in the convex body $F(K)$ of volume $1$.
Since $F$ translates the uniform distribution on $F(K)$ to the uniform distribution on $K$ and preserves holes and islands, we obtain the required upper bound also in the general case.
\end{proof}

Now, we finish the proof of Theorem~\ref{thm:islands_2d}.

\begin{proof}[Proof of Theorem~\ref{thm:islands_2d}]
We estimate the expected value of the number $X$ of $k$-islands in $S$.
The number of ordered $k$-tuples of points from $S$ is $n(n-1)\cdots(n-k-1)$.
Since every subset of $S$ of size $k$ admits a unique labeling that satisfies the conditions~\ref{item-canonical0}, ~\ref{item-canonical1}, \ref{item-canonical4}, \ref{item-canonical2}, and \ref{item-canonical3}, we have
\begin{align*}
\mathbb{E}[X] &= n(n-1)\cdots(n-k+1) \cdot \Pr\left[\cup_{a=0}^{k-d-1} E_{a,k}\right] \\
&= n(n-1)\cdots(n-k+1) \cdot \sum_{a=0}^{k-d-1} \Pr\left[E_{a,k}\right],
\end{align*}
as the events $E_{0,k},\dots E_{k-d-1,k}$ are pairwise disjoint.

The probability of the event $L_2$, which says that the points $p_1,\dots,p_d$ satisfy the condition~\ref{item-canonical1}, is $1/d!$.
Let $P = \sum_{a=0}^{k-d-1} \Pr\left[E_{a,k} \mid L_2 \right]$.
For any two events $E,E'$ with $E \supseteq E'$ and $\Pr[E]>0$, we have $\Pr[E'] = \Pr[E \cap E'] = \Pr[E' \mid E]\cdot \Pr[E]$.
Thus, using $L_2 \supseteq E_a = E_{a,d+a+1} \supseteq E_{a,d+a+2} \supseteq \cdots \supseteq E_{a,k}$, we get
\[ \mathbb{E}[X] = n(n-1)\cdots(n-k+1) \cdot \Pr[L_2] \cdot P = \frac{n(n-1)\cdots(n-k+1)}{d!} \cdot P\]
and
\[
P = \sum_{a=0}^{k-d-1}\Pr[E_a \mid L_2]\cdot \prod_{i=d+a+2}^k \Pr[E_{a,i} \mid E_{a,i-1}].
\]

For every $a \in \{d+2,\dots,k-d-1\}$, Lemma~\ref{lem-probabilityInside} gives
\[\Pr[E_a \mid L_2] \leq \frac{2^{d-1}\cdot d!}{(k-a-d-1)!\cdot(n-k+1)^{a+1}} \leq \frac{2^{d-1}\cdot d!}{(n-k+1)^{a+1}}\]
and, due to Lemma~\ref{lem-probabilityOutside},
\[\Pr[E_{a,i} \mid E_{a,i-1}] \leq \frac{2d^{2d-1} \cdot  \binom{k}{\lfloor d/2 \rfloor}}{n-i+1}\]
for every $i \in \{d+a+2,\dots,k\}$.

Using these estimates we derive
\begin{align*}
P &\leq 2^{d-1}\cdot d! \cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot \sum_{a=0}^{k-d-1}\frac{1}{(n-k+1)^{a+1}} \cdot \prod_{i=d+a+2}^k \frac{1}{n-i+1}\\
& \leq 2^{d-1}\cdot d! \cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot \sum_{a=0}^{k-d-1}\frac{1}{(n-k+1)^{a+1}} \cdot \frac{1}{(n-k+1)^{k-d-a-1}}\\
&= 2^{d-1}\cdot d! \cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot (k-d) \cdot \frac{1}{(n-k+1)^{k-d}}.
\end{align*}

Thus the expected number of $k$-islands in $S$ satisfies
\begin{align*}
\mathbb{E}[X] &= \frac{n(n-1) \cdots (n-k+1)}{d!} \cdot P\\
&\leq \frac{2^{d-1}\cdot d! \cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot (k-d)}{d!} \cdot \frac{n(n-1) \cdots (n-k+1)}{(n-k+1)^{k-d}} \\
&=2^{d-1}\cdot \left(2d^{2d-1}\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} \cdot (k-d) \cdot \frac{n(n-1) \cdots (n-k+2)}{(n-k+1)^{k-d-1}}.
\end{align*}
This finishes the proof of Theorem~\ref{thm:islands_2d}.
\end{proof}

In the rest of the section, we sketch the proof of Theorem~\ref{thm:holes_2d} by showing that a slight modification of the above proof yields an improved bound on the expected number $EH^K_{d,k}(n)$ of $k$-holes in $S$.

\begin{proof}[Sketch of the proof of Theorem~\ref{thm:holes_2d}]
If $k$ points from $S$ determine a $k$-hole in $S$, then, in particular, the simplex $\Delta$ contains no points of $S$ in its interior.
Therefore
\[EH^K_{d,k}(n) \leq n(n-1)\cdots(n-k+1) \cdot \Pr[E_{0,k}].\]
Then we proceed exactly as in the proof of Theorem~\ref{thm:islands_2d}, but we only consider the case $a=0$.
This gives the same bounds as before with the term $(k-d)$ missing and with an additional factor $\frac{1}{(k-d-1)!}$ from Lemma~9, which proves Theorem~\ref{thm:holes_2d}.
\end{proof}

For $d=2$ and $k=4$, Theorem~\ref{thm:holes_2d} gives $EH^K_{2,4}(n) \leq 128n^2 + o(n^2)$.
We can obtain an even better estimate $EH^K_{2,4}(n) \leq 12n^2 + o(n^2)$ in this case.
First, we have only three facets $\varphi$, as they correspond to the sides of the triangle $\Delta$.
Thus the term $\left(2\binom{k}{\lfloor d/2 \rfloor}\right)^{k-d-1} = 8$ is replaced by $3$.
Moreover, the inequality~\eqref{eq-simplex} can be replaced by
\[\lambda_1(C_\varphi \cap H(h) \cap \Delta^*) \leq \lambda_1(\varphi),\]
since every line $H(h)$ intersects $R_j \subseteq \Delta^*$ in a line segment of length at most $\lambda_1(F_j) = \lambda(\varphi)$.
This then removes the factor $d^{(2d-2)(k-d-1)} = 4$.

\section{Proof of Theorem~\ref{thm:exponential}}
\label{sec:exponentially_k_islands}

For an integer $d \geq 2$ and a convex body $K$ in $\mathbb{R}^d$ with $\lambda_d(K)=1$.
We show that there are positive constants $C_1=C_1(d)$, $C_2=C_2(d)$, and $n_0=n_0(d)$ such that for every integer $n \geq n_0$ the expected number of islands in a set $S$ of $n$ points chosen uniformly and independently at random from $K$ is at least $2^{C_1 \cdot n^{(d-1)/(d+1)}}$ and at most $2^{C_2 \cdot n^{(d-1)/(d+1)}}$.

For every point set $Q$, there is a one-to-one correspondence between the set of all subsets in $Q$ in convex position and the set of all islands in $Q$.
It suffices to map a subset $G$ of $Q$ in convex position to the island $Q \cap \conv(G)$ in~$Q$.
On the other hand, every island $I$ in $Q$ determines a subset of $Q$ in convex position that is formed by the vertices of $\conv(I)$.
Therefore the number of subsets of $Q$ in convex position equals the number of islands in~$Q$.

For $m \in \mathbb{N}$, let $p(m,K)$ be the probability that $m$ points chosen uniformly and independently at random from $K$ are in convex position.
We use the following result by B\'{a}r\'{a}ny~\cite{barany01} to estimate the expected number of islands in $S$.

\begin{theorem}[\cite{barany01}]
\label{thm-barany}
For every integer $d \geq 2$, let $K$ be a convex body in $\mathbb{R}^d$ with $\lambda_d(K)=1$.
Then there are positive constants $m_0$, $c_1$, and $c_2$ depending only on $d$ such that, for every $m \geq m_0$, we have
\[c_1 < m^{2/(d-1)} \cdot (p(m,K))^{1/m} < c_2.\]
\end{theorem}

We let $X$ be the random variable that denotes the number of subsets of $S$ in convex position.
Then
\[\mathbb{E}[X] = \sum_{m=1}^n\binom{n}{m} \cdot p(m,K).\]

Now, we prove the upper bound on the expected number of islands in $S$.
By Theorem~\ref{thm-barany}, we have
\[\mathbb{E}[X] \leq \sum_{m=1}^{m_0-1}\binom{n}{m} + \sum_{m=m_0}^n\binom{n}{m} \cdot \left(\frac{c_2}{m^{2/(d-1)}}\right)^m\]
for some constants $m_0$ and $c_2$ depending only on $d$.
Since $\binom{n}{m} \leq n^m$, the first term on the right side is at most $n^c$ for some constant $c=c(d)$.
Using the bound $\binom{n}{m} \leq (en/m)^m$, we bound the second term from above by
\[\sum_{m=m_0}^n\ \left(\frac{c_2 \cdot e\cdot n}{m^{(d+1)/(d-1)}}\right)^m.\]
The real function $f(x)=(e \cdot c_2 \cdot n/x^{(d+1)/(d-1)})^x$ is at most $1$ for $x \geq (e\cdot c_2 \cdot n)^{(d-1)/(d+1)}$.
Otherwise $x=y (e\cdot c_2 \cdot n)^{(d-1)/(d+1)}$ for some $y \in [0,1]$ and
\[f(x) = \left(y^{-(d+1)/(d-1)}\right)^{y(e\cdot c_2 \cdot n)^{(d-1)/(d+1)}} = e^{\frac{d+1}{d-1}y\ln{(1/y)}(e\cdot c_2 \cdot n)^{(d-1)/(d+1)}} \leq 2^{c'n^{(d-1)/(d+1)}},\]
where $c'=c'(d)$ is a sufficiently large constant.
Thus $\mathbb{E}[X] \leq n^c + n2^{c'n^{(d-1)/(d+1)}}$.
Choosing $C_2=C_2(d)$ sufficiently large, we have $\mathbb{E}[X] \leq 2^{C_2n^{(d-1)/(d+1)}}$.
Since the number of subsets of $S$ in convex position equals the number of islands in $S$, we have the same upper bound on the expected number of islands in $S$.

It remains to show the lower bound on the expected number of islands in $S$.
By Theorem~\ref{thm-barany}, we have
\[\mathbb{E}[X] \geq \sum_{m=m_0}^n\binom{n}{m} \cdot \left(\frac{c_1}{m^{2/(d-1)}}\right)^m\]
for some constants $m_0$ and $c_1$ depending only on $d$.
Using the bound $\binom{n}{m} \geq (n/m)^m$, we obtain
\[\mathbb{E}[X] \geq \sum_{m=m_0}^n\cdot \left(\frac{c_1 \cdot n}{m^{(d+1)/(d-1)}}\right)^m.\]
Let $C_1=(c_1/2)^{(d-1)/(d+1)}$.
Then, for each $m$ satisfying $C_1 n^{\frac{d-1}{d+1}}/2 \leq m \leq C_1 n^{\frac{d-1}{d+1}}$, the summand in the above expression is at least $2^{C_1 n^{\frac{d-1}{d+1}}/2}$.
It follows that the expected number of $m$-tuples from $S$ in convex position, where $C_1 n^{\frac{d-1}{d+1}}/2 \leq m \leq C_1 n^{\frac{d-1}{d+1}}$, is at least
\[2^{C_1 n^{(d-1)/(d+1)}/2},\]
as if $n$ is large enough with respect to $d$, then there is at least one $m \in \mathbb{N}$ in the given range.
As we know, each such $m$-tuple in $S$ corresponds to an island in $S$.
Thus the expected number of islands in $S$ is also at least $2^{C_1 n^{(d-1)/(d+1)}/2}$.

\section{Proof of Theorem~\ref{prop-Horton}}
\label{sec:prop-Horton}

Here, for every $d$, we state the definition of a $d$-dimensional analogue of Horton sets on $n$ points from~\cite{VALTR1992b} and show that, for all fixed integers $d$ and $k$, every $d$-dimensional Horton set $H$ with $n$ points contains at least $\Omega(n^{\min\{2^{d-1},k\}})$ $k$-islands in $H$.
If $k \leq 3 \cdot 2^{d-1}$, then we show that $H$ contains at least $\Omega(n^k)$ $k$-holes in $H$.

First, we need to introduce some notation.
A set $Q$ of points in $\mathbb{R}^d$ is in \emph{strongly general position} if $Q$ is in general position and, for every $i=1,\dots,d-1$, no $(i+1)$-tuple of points from $Q$ determines an $i$-dimensional affine subspace of $\mathbb{R}^d$ that is parallel to the $(d-i)$-dimensional linear subspace of $\mathbb{R}^d$ that contains the last $d-i$ axes.
Let $\pi \colon \mathbb{R}^d \to \mathbb{R}^{d-1}$ be the projection defined by $\pi(x_1,\dots,x_d) = (x_1,\dots,x_{d-1})$.
For $Q \subseteq \mathbb{R}^d$, we use $\pi(Q)$ to denote the set $\{\pi(q) \in \mathbb{R}^{d-1} \colon q \in Q\}$.
If $Q$ is a set of $n$ points $q_0,\dots,q_{n-1}$ from $\mathbb{R}^d$ in strongly general position that are ordered so that their first coordinates increase, then, for all $m \in \mathbb{N}$ and $i \in \{0,1,\dots,m-1\}$, we define $Q_{i,m}=\{q_j \in Q \colon j \equiv i \; (\bmod \; m)\}$.

For two sets $A$ and $B$ of points from $\mathbb{R}^d$ with $|A|,|B| \geq d$, we say that $B$ is \emph{deep below} $A$ and $A$ is \emph{high above} $B$ if $B$ lies entirely below any hyperplane determined by $d$ points of $A$ and $A$ lies entirely above any hyperplane determined by $d$ points of $A$.
For point sets $A'$ and $B'$ in $\mathbb{R}^d$ of arbitrarily size, we say that $B'$ is \emph{deep below} $A'$ and $A'$ is \emph{high above} $B'$ if there are sets $A \supseteq A'$ and $B \supseteq B'$ such that $|A|,|B| \geq d$, $B$ is deep below $A$, and $A$ is high above~$B$.

Let $p_2<p_3<p_4<\cdots$ be the sequence of all prime numbers.
That is, $p_2=2$, $p_3=3$, $p_4=5$ and so on.

We can now state the definition of the $d$-dimensional Horton sets from~\cite{VALTR1992b}.
Every finite set of $n$ points in $\mathbb{R}$ is \emph{$1$-Horton}.
For $d \geq 2$, finite set $H$ of points from $\mathbb{R}^d$ in strongly general position is a \emph{$d$-Horton set} if it satisfies the following conditions:
\begin{enumerate}[label=(\alph*)]
\item the set $H$ is empty or it consists of a single point, or
\item $H$ satisfies the following three conditions:
\begin{enumerate}[label=(\roman*)]
\item if $d>2$, then $\pi(H)$ is $(d-1)$-Horton,
\item for every $i \in \{0,1,\dots,p_d-1\}$, the set $H_{i,p_d}$ is $d$-Horton,
\item every $I \subseteq \{0,1,\dots,p_d-1\}$ with $|I| \geq 2$ can be partitioned into two nonempty subsets $J$ and $I \setminus J$ such that $\cup_{j \in J}H_{j,p_d}$ lies deep below $\cup_{i \in I \setminus J} H_{i,p_d}$.
\end{enumerate}
\end{enumerate}

Valtr~\cite{VALTR1992b} showed that such sets indeed exist and that they contain no $k$-hole with $k>2^{d-1}(p_2p_3\cdots p_d+1)$.
The $2$-Horton sets are known as \emph{Horton sets}.
We show that $d$-Horton sets with $d \geq 3$ contain many $k$-islands for $k\geq d+1$ and thus cannot provide the upper bound $O(n^d)$ that follows from Theorem~\ref{thm:islands_2d}.
This contrasts with the situation in the plane, as 2-Horton sets of $n$ points contain only $O(n^2)$ $k$-islands for any fixed~$k$~\cite{FabilaMonroyHuemer2012}.

Let $d$ and $k$ be fixed positive integers.
Assume first that $k \geq 2^{d-1}$.
We want to prove that there are $\Omega(n^{2^{d-1}})$ $k$-islands in every $d$-Horton set $H$ with $n$ points.
We proceed by induction on $d$.
For $d=1$ there are $n-k+1=\Omega(n)$ $k$-islands in every $1$-Horton set.

Assume now that $d>1$ and that the statement holds for $d-1$.
The $d$-Horton set $H$ consists of $p_d \in O(1)$ subsets $H_{i,p_d}$, each of size at least $\lfloor n/p_d \rfloor \in \Omega(n)$.
The set $\{0,\dots,p_d-1\}$ is ordered by a linear ordering $\prec$ such that, for all $i$ and $j$ with $i \prec j$, the set $H_{i,p_d}$ is deep below $H_{j,p_d}$; see~\cite{VALTR1992b}.
Take two of sets $X = H_{a,p_d}$ and $Y = H_{b,p_d}$ such that $a \prec b$ are consecutive in $\prec$.
Since $k \geq 2^{d-1}$, we have $\lceil k/2 \rceil \geq  \lfloor k/2 \rfloor \geq 2^{d-2}$.
Thus by the inductive hypothesis, the $(d-1)$-Horton set $\pi(X)$ of size at least $\Omega(n)$ contains at least $\Omega(n^{2^{d-2}})$ $\lfloor k/2 \rfloor$-islands.
Similarly, the $(d-1)$-Horton set $\pi(Y)$ of size at least $\Omega(n)$ contains at least $\Omega(n^{2^{d-2}})$ $\lceil k/2 \rceil$-islands.

Let $\pi(A)$ be any of the $\Omega(n^{2^{d-2}})$ $\lfloor k/2 \rfloor$-islands in $\pi(X)$, where $A\subseteq X$.
Similarly, let $\pi(B)$ be any of the $\Omega(n^{ 2^{d-2} })$
$\lceil k/2 \rceil$-islands in $\pi(Y)$, where $B\subseteq Y$.
We show that $A \cup B$ is a $k$-island in $H$.
Suppose for contradiction that there is a point $x \in H \setminus (A \cup B)$ that lies in $\conv(A \cup B)$.
Since $a$ and $b$ are consecutive in $\prec$, the point $x$ lies in $X \cup Y = H_{a,p_d} \cup H_{b,p_d}$.
By symmetry, we may assume without loss of generality that $x \in X$.
Since $x \notin A$ and $H$ is in strongly general position, we have $\pi(x) \in \pi(X) \setminus \pi(A)$.
Using the fact that $\pi(A)$ is a $\lfloor k/2 \rfloor$-island in $\pi(X)$, we obtain $\pi(x) \notin \conv(\pi(A))$ and thus $x \notin \conv(A)$.
Since $X$ is deep below $Y$, we have $x \notin \conv(B)$.
Thus, by Carath\'{e}dory's theorem, $x$ lies in the convex hull of a $(d+1)$-tuple $T \subseteq A \cup B$ that contains a point from~$A$ and also a point from~$B$.

Note that, for $U = (T \cup \{x\})$, we have $|U \cap A| \geq 2$, as $x \in A$ and $|T \cap A| \geq 1$.
We also have $|U \cap B| \geq 2$, as $X$ is deep below $Y$ and $\pi(x) \notin \conv(\pi(A))$.
Thus the affine hull of $U \cap A$ intersects the convex hull of $U \cap B$.
Then, however, the set $U \cap A$ is not deep below the set $U \cap B$, which contradicts the fact that $X$ is deep below~$Y$.

Altogether, there are at least $\Omega(n^{ 2^{d-2} })\cdot\Omega(n^{ 2^{d-2}
})=\Omega(n^{ 2^{d-1} })$ such $k$-islands $A\cup B$, which finishes the proof if $k$ is at least $2^{d-1}$.
For $k<2^{d-1}$, we use an analogous argument that gives at least $\Omega(n^{\lfloor k/2 \rfloor}) \cdot \Omega(n^{\lceil k/2 \rceil}) = \Omega(n^k)$ $k$-islands in the inductive step.

If $d \geq 2$ and $k \leq 3 \cdot 2^{d-1}$ then a slight modification of the above proof gives $\Omega(n^{\min\{2^{d-1},k\}})$ $k$-islands which are actually $k$-holes in $H$.
We just use the simple fact that every 2-Horton set with $n$ points contains $\Omega(n^2)$ $k$-holes for every $k \in \{2,\dots,6\}$ as our inductive hypothesis.
This is trivial for $k=2$ and it follows for $k \in \{3,4\}$ from the well-known fact that every set of $n$ points in $\mathbb{R}^2$ in general position contains at least $\Omega(n^2)$ $k$-holes.
For $k \in \{5,6\}$, this fact can be proved using basic properties of 2-Horton sets (we omit the details).
Then we use the inductive assumption, which says that every $d$-Horton set of $n$ points contains at least $\Omega(n^{\min\{2^{d-1},k\}})$ $k$-holes if $d \geq 2$ and $1 \leq k \leq 3 \cdot 2^{d-1}$.
This finishes the proof of Theorem~\ref{prop-Horton}.