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 Snβ1 with
a geodesic distance 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. We also prove that in
general probability metric spaces the sup in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page