Brunn–Minkowski inequalities in product metric measure spaces

Abstract

Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p≥−1, we prove that the product X×Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1/(1+p−1) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X×Y when Y satisfies a Prékopa–Leindler inequality.In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in Rn (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality.Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results.