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

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:

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!

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!