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 $E$. We assume that $C$ is a closed convex set of $E$ such that $C \cap X(\Omega)$ contains at least two elements. Handle the case $n = 1$.