A Gaussian small deviation inequality for convex functions

Abstract

Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that

\[\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),\] for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.