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.
Highlights
Let us recall a well known result for Lipschitz functions on probability metric spaces,(V, d, μ)
Where F is the complete class of integrable, Lipschitz functions on probability metric spaces
As corollaries we get exact solutions of (0.1) for Euclidean unit sphere Sn−1 with a geodesic distance function and a normalized Haar measure, for Rn equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance function
Summary
Let us recall a well known result for Lipschitz functions on probability metric spaces,(V, d, μ). Where F is the complete class of integrable, Lipschitz functions on probability metric (product) spaces. We prove that in general probability metric spaces the sup in (0.1) is achieved on a family of negative distance functions.
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