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

> <cite authors="Talagrand, M.">_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).</cite>

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 $||x||_C := \inf \{ \lambda : x \in \lambda C \}$. 

**Lemma 1.**
For each $\varepsilon \in (0,1)$, there is a universal $R(\varepsilon)$ with the following property. For any $m$ and any 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 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:

> <cite authors="Talagrand, Michel">_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).</cite>

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!