I'm looking for references for two facts that are stated without proof in the paper:



> 
> *Talagrand, M.*, Are all sets of
> positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
> Geometric aspects of functional analysis. Israel seminar (GAFA)
> 1992-94. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 77, 294-310
> (1995). [ZBL0835.60003](https://zbmath.org/?q=an:0835.60003).
> 
> 
> 


Let $\gamma\_m(ds) := (2\pi)^{-m/2}e^{-||s||\_2^2/2}ds$ be the standard Gaussian measure on $\mathbb{R}^m$. Given a convex set $C$, define the guage $||x||\_C := \inf \{ \lambda : x \in \lambda C \}$. A convex set is balanced if $s \in \mathbb{C}$ implies $\lambda s \in \mathbb{C}$ for every $\lambda \in [-1,1]$.


**Lemma 1.**
For each $\varepsilon \in (0,1)$, there is a universal $R(\varepsilon)$ with the following property. For any $m$ and any balanced convex $C \subset \mathbb{R}^m$ satisfying $\gamma\_m(C) > \varepsilon$, we have $\int\_{\mathbb{R}^m}||s||\_C \gamma\_m(ds) \leq R(\varepsilon)$.


We say that an $\mathbb{R}^m$-valued random variable $Y = (Y\_1,\ldots,Y\_m)$ is subgaussian if $\mathbb{E}[e^{ \sum\_{i=1}^m a\_i Y\_i } ] \leq e^{ \frac{1}{2} \sum\_{i=1}^m a\_i^2} $ for all $a\_1,\ldots,a\_m \in \mathbb{R}$. We have the following generalisation:


**Lemma 2.**
For each $\varepsilon \in (0,1)$, there is a universal $S(\varepsilon)$ with the following property. For any $m$ and any balanced convex set $C$ satisfying $\gamma\_m(C) > \varepsilon$, and any $Y$ subgaussian, we have $\mathbb{E}[||Y||\_C] \leq S(\varepsilon)$.


Talagrand implies that the latter lemma follows from results in another paper of his:



> 
> *Talagrand, Michel*, [**Regularity
> of Gaussian processes**](https://doi.org/10.1007/BF02392556), Acta
> Math. 159, No. 1-2, 99-149 (1987).
> [ZBL0712.60044](https://zbmath.org/?q=an:0712.60044).
> 
> 
> 


However, I wasn't able to find a statement like Lemma 2 in the latter.


I'd be very grateful if someone could point me to a reference, or even a proof!



Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
$\newcommand{\R}{\mathbb R}\renewcommand{\th}{\theta}\newcommand{\ep}{\varepsilon} $This indeed follows immediately from [Theorem 15](https://doi.org/10.1007/BF02392556) of Talagrand, Regularity of Gaussian processes, Acta Math. 159, No. 1-2, 99-149 (1987).




---


Details: The mentioned theorem by Talagrand states the following:



> 
> Let $(X\_t)\_{t\in T}$ be a Gaussian process, and $(Y\_t)\_{t\in T}$ be any other centered
> process indexed by the same set. Assume that for each $\th\in\R$, we have
> \begin{equation}
> E \exp \th(Y\_u-Y\_v)\le E \exp \th(X\_u-X\_v) = \exp(\th^2 d(u, v)^2/2). \quad (40)
> \end{equation}
> Then we have $E\sup\_{t\in T}Y\_t\le K E\sup\_{t\in T}X\_t$ where $K$ is a universal constant.
> 
> 
> 


Let now $T$ be a set of vectors $t=(t\_1,\dots,t\_m)\in\R^m$ such that $\|x\|\_C=\sup\_{t\in T}t\cdot x$ for all $x\in\R^m$, where $\cdot$ denotes the dot product. For $t=(t\_1,\dots,t\_m)\in\R^m$, let $Y\_t:=t\cdot Y$ and $X\_t:=t\cdot X$, where $X$ is a standard normal random vector in $\R^m$. Then (40) will hold with $d(u,v)^2=(u-v)\cdot(u-v)$. So, we will have $E\|Y\|\_C\le K E\|X\|\_C$. So, one may take $S(\ep):=KR(\ep)$. $\quad\Box$


