# Brunn-Minkowski inequalities in product metric measure spaces

## TL;DR

This paper extends Brunn-Minkowski inequalities to product metric measure spaces, establishing new inequalities under mild conditions and applying them to weakly unconditional sets in Euclidean space, with implications for isoperimetric problems.

## Contribution

It proves new Brunn-Minkowski inequalities in product spaces and applies them to weakly unconditional sets, improving understanding of Gaussian isoperimetric inequalities.

## Key findings

- Product space Brunn-Minkowski inequality of order 1/(1+p^{-1})
- Class of weakly unconditional sets satisfies classical Brunn-Minkowski inequality
- Derived isoperimetric inequalities from the new Brunn-Minkowski results

## 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 Pr\'ekopa-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.