Let $\gamma$ be a positive numerical constant. We assume the following property is satisfied: for every convex set $A \subset Q^N$, $$\mathbb{P}(X \in A) \mathbb{E}\left[\exp\left(\gamma \frac{d(X,A)^2}{4K^2}\right)\right] \leq 1.$$ We are given a 1-Lipschitz and convex function $F : \mathbb{R}^N \rightarrow \mathbb{R}$. The purpose of this question is to prove that (37) is then verified: $$\mathbb{P}(F(X) \geq m) \geq \frac{1}{2} \Longrightarrow \mathbb{P}(F(X) \leq m - t) \leq \beta e^{-\alpha t^2/K^2}$$ II.1.a) Let $s, \sigma \in \mathbb{R}$ with $s < \sigma$. By considering the set $$A_s = \left\{x \in Q^N; F(x) \leq s\right\}$$ show that $$\mathbb{P}(F(X) \leq s)\mathbb{P}(F(X) \geq \sigma) \leq \exp\left(-\gamma \frac{(\sigma - s)^2}{4K^2}\right)$$ II.1.b) Prove that (37) is verified.
Let $\gamma$ be a positive numerical constant. We assume the following property is satisfied: for every convex set $A \subset Q^N$,
$$\mathbb{P}(X \in A) \mathbb{E}\left[\exp\left(\gamma \frac{d(X,A)^2}{4K^2}\right)\right] \leq 1.$$
We are given a 1-Lipschitz and convex function $F : \mathbb{R}^N \rightarrow \mathbb{R}$. The purpose of this question is to prove that (37) is then verified:
$$\mathbb{P}(F(X) \geq m) \geq \frac{1}{2} \Longrightarrow \mathbb{P}(F(X) \leq m - t) \leq \beta e^{-\alpha t^2/K^2}$$
\textbf{II.1.a)} Let $s, \sigma \in \mathbb{R}$ with $s < \sigma$. By considering the set
$$A_s = \left\{x \in Q^N; F(x) \leq s\right\}$$
show that
$$\mathbb{P}(F(X) \leq s)\mathbb{P}(F(X) \geq \sigma) \leq \exp\left(-\gamma \frac{(\sigma - s)^2}{4K^2}\right)$$
\textbf{II.1.b)} Prove that (37) is verified.