Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand

Question

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.

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, Acta Math. 159, No. 1-2, 99-149 (1987). ZBL0712.60044.

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!

Answer 1

$\newcommand{\R}{\mathbb R}\renewcommand{\th}{\theta}\newcommand{\ep}{\varepsilon} $This indeed follows immediately from Theorem 15 of Talagrand, Regularity of Gaussian processes, Acta Math. 159, No. 1-2, 99-149 (1987).

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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 $S(\ep):=KR(\ep)$. $\quad\Box$