# Gaussian vectors, half-spaces, and convexity

Let A be a subset of R^n and let B be a half-space with the same Gaussian measure as A. For a pair of correlated Gaussian vectors X and Y, Pr(X \in A, Y \in A) is smaller than Pr(X \in B, Y \in B); this was originally proved by Borell, who also showed various other extremal properties of half-spaces. For example, the exit time of an Ornstein-Uhlenbeck process from A is stochastically dominated by its exit time from B. We will discuss these (and other) inequalities using a kind of modified convexity.

This talk is part of the Statistics series.