Extremal Lipschitz functions in the deviation inequalities from the mean∗

We obtain an optimal deviation from the mean upper bound D(x) def = sup f∈F μ{f −Eμf ≥ x}, for x ∈ R (0.1) where F is the complete class of integrable, Lipschitz functions on probability metric (product) 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 R equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance function. We also prove that in general probability metric spaces the sup in (0.1) is achieved on a family of negative distance functions.