Extremal Lipschitz functions in the deviation inequalities from the mean

TL;DR

This paper derives optimal deviation bounds for Lipschitz functions on various metric probability spaces, providing exact solutions for specific geometries and showing the supremum is achieved on a family of distance functions.

Contribution

It introduces the first optimal deviation bounds for Lipschitz functions in general metric probability spaces, with explicit solutions for classical geometries.

Findings

Exact deviation bounds for Euclidean spheres, Gaussian spaces, and discrete structures.

The supremum in the deviation inequality is achieved on a family of distance functions.

Provides a unified approach to deviation inequalities across different metric spaces.

Abstract

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of $\eqref{abstr}$ for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ 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 $\sup$ in $\eqref{abstr}$ is achieved on a family of distance functions.