Title: 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 $p\ge -1$, we prove that the product $X\times Y$ 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 $X\times Y$ 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 $\mathbb{R}^n$ (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.