Gaussian concentration in the Kantorovich metric of distributions of random variables and the quantile functions

A sketch of the proof of the following theorem. Let the unit ball of the kernel space $H_\gamma$ of a centered Gaussian measure $\gamma$ in the space $L^2$ is a subspace of the unit ball of this space. There exists a (“typical”) univariate distribution $\bar{\mathbf P}_\gamma$ such that the expectation with respect to $\gamma$ of the Kantorovich distance between the distribution of an element of $L^2$ chosen at random and this typical distribution is less than 0.8.