Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that 
\begin{equation}
\int\_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and assume that $f$ has a strict global maximum $x\_0$, that is $f(x) < f(x\_0)$ for all $x \neq x\_0$.
For any fixed $q \in (0,1]$ consider the problem
\begin{equation}
\min\_{y \in \mathbb{R}} \int\_{\mathbb{R}}|y-x|^{q} f(x) \,dx,
\end{equation}


and let $S\_q$ be the set of all minimizers for this problem. My question is: does $S\_q$ "shrink" to $x\_0$ for $q \rightarrow 0$, in the sense that for every $\epsilon > 0$ there is some $\delta > 0$ such that $S\_q \subset (x\_0 - \epsilon, x\_0 + \epsilon)$ for every $q \in (0, \delta)$?


I found this last property stated in a monograph about applied statistics without any proof, but I am not so convinced of its validity.


NOTE (1). If we put for a fixed $q \in (0,1]$:
\begin{equation}
F\_q(y)=\int\_{\mathbb{R}} |y-x|^q f(x) dx \quad (y \in \mathbb{R}),
\end{equation}
then $F\_q$ is a continuous function and $\lim\_{y \rightarrow - \infty} F\_q(y) = \lim\_{y \rightarrow \infty} F\_q(y) = \infty$, so the minimization problem 


\begin{equation}
\min\_{y \in \mathbb{R}} \int\_{\mathbb{R}}|y-x|^{q} f(x) \,dx.
\end{equation}


admits for sure some solution. 


NOTE (2). For every $q > 0$ let $g\_q:\mathbb{R} \rightarrow \mathbb{R}$ be the function:
\begin{equation}
g\_q(x) = 
 \begin{cases}
 1 &\quad &\text{if} &\quad |x| > q,\\
 0 &\quad &\text{if} &\quad |x| \leq q,
 \end{cases}
\end{equation}
and let us put
\begin{equation}
G\_q(y)=\int\_{\mathbb{R}} g\_q(y-x)f(x) dx.
\end{equation}
We have $G\_q \leq 1$, $G\_q(y) < 1$ for some $y \in \mathbb{R}$ and $\lim\_{y \rightarrow - \infty} G\_q(y) = \lim\_{y \rightarrow \infty} G\_q(y) = 1$, so that the problem
\begin{equation}
\min\_{y \in \mathbb{R}} G\_q(y)
\end{equation}
admits a non-empty compact set $T\_q$ of minimizers. Moreover it is immediate to see arguing by contradiction that $T\_q$ shrinks to $x\_0$ as $q \rightarrow 0$ (in the precise sense explained above). This result might have suggested the conjecture about $S\_q$ of our original problem.



q-Means and the mode of a distribution
Finally, I discovered that this problem was deeply investigated by the great french mathematician Maurice Fréchet, who introduced in his remarkable essay [Les éléments aléatoires de nature quelconque dans un espace distancié](http://www.numdam.org/article/AIHP_1948__10_4_215_0.pdf) the general notion of q-mean, where $q$ is any positive real number, as a minimizer of
\begin{equation}
F\_q(y)=\int\_{\mathbb{R}} |y-x|^q f(x) dx \quad (y \in \mathbb{R}).
\end{equation}
Let us note that this objective function can be written in several interesting ways. First of all $F\_q$ is simply the convolution of $f$ with the map $x \mapsto |x|^q$. Second, we have $F\_q(y)= \mathbb{E} [|X-y|^q]$, where $X$ is a random variable with density function $f$. Finally, for $q \geq 1$ we also have $F\_q(y)= || \mathbb{1} - y ||^{q}\_q$ where $\mathbb{1}$ is the identity function on $\mathbb{R}$ and $||.||\_q$ is the norm of the space $L^q(\mathbb{R},\mathcal{B}(\mathbb{R}), \mu)$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of $\mathbb{R}$ and $\mu$ the probability measure defined by $f$:
\begin{equation}
\mu(A)=\int\_{A} f(x)dx \quad (A \in \mathcal{B}(\mathbb{R})).
\end{equation}
The notion of q-mean was extensively studied by Fréchet in his following essay [Positions typiques d'un élément aléatoire de nature quelconque](http://www.numdam.org/article/ASENS_1948_3_65__211_0.pdf), while he dedicated an entire work [Les valeurs typiques d'ordre nul ou infini d'un nombre aléatoire](https://www.jstor.org/stable/1401229?seq=1#page_scan_tab_contents) to study the limit cases $q=0$ and $q =\infty$. 


In this last work , he introduces three different definitions for the case $q=0$, with the third one directly related to our conjecture. With the notation already introduced this last definition is the following.


$\textbf{Definition}$. A number $x \in \mathbb{R}$ is said to be a 0-mean (or mean of order zero) of the distribution defined by $f$ if there exists a sequence of positive numbers $(q\_n)$ such that $\lim\_{n \rightarrow 0} q\_n=0$ and $\lim\_{n \rightarrow 0} x\_{q\_n}= x$, where $x\_{q\_n} \in S\_{q\_n}$ for every $n$.


Then Fréchet finds a counterexample (discussed at pages 16-19 of the work) of a strongly unimodal $f$ (that is an $f$ which is non-decreasing on $(-\infty,x\_0]$ and non-increasing on $[x\_0,\infty)$) for which the 0-mean is unique and does not coincide with the mode. In particular, this implies that the conjecture of the post is false, as I suspected. Even though the $f$ in the counterexample found by Fréchet is very simple, the proof is not trivial at all and shows all the incredible analytical ability of this mathematical genius. 


