Extremal Lipschitz functions in the deviation inequalities from the mean

Original · EN

We obtain an optimal deviation from the mean upper bound equation D(x)∈ μ{f-μ f≥ x},forx∈abstr equation where is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of abstr for Euclidean unit sphere Sⁿ⁻¹ with a geodesic distance and a normalized Haar measure, for 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 in abstr is achieved on a family of distance functions.