Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ such that $C \cap X(\Omega)$ contains at least two elements. We propose to prove inequality $$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$ by induction on the dimension $n$ of $E$. Handle the case $n = 1$.
Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ such that $C \cap X(\Omega)$ contains at least two elements. We propose to prove inequality
$$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$
by induction on the dimension $n$ of $E$. Handle the case $n = 1$.