Extremal Lipschitz functions in the deviation inequalities from the mean

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.