Extremal Lipschitz functions in the deviation inequalities from the mean

Abstract

We obtain an optimal deviation from the mean upper bound $D(x)=\sup\{\mu\{f-\mathbb{E}_{\mu} f\geq x\}:f\in\mathcal{F},x\in\mathbb{R}\}$ where $\mathcal{F}$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact bounds for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\mathbb{R}^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove that in general probability metric spaces the $\sup$ is achieved on a family of distance functions.