A note on an $L^p$-Brunn-Minkowski inequality for convex measures in the unconditional case

TL;DR

This paper establishes an $L^p$-Brunn-Minkowski inequality for convex measures under unconditional assumptions, extending previous results and exploring concavity properties of measure functions, with implications for the (B)-conjecture.

Contribution

It introduces a new $L^p$-Minkowski combination and proves an inequality for convex measures, including log-concave measures, in the unconditional setting.

Findings

Proved an $L^p$-Brunn-Minkowski inequality for convex measures with $p o [0,1]$

Derived concavity properties of measure functions for unconditional convex bodies

Showed the equivalence of the (B)-conjecture for uniform and log-concave measures.

Abstract

We consider a different $L^p$-Minkowski combination of compact sets in $\mathbb{R}^n$ than the one introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function $t \mapsto \mu(t^{\frac{1}{p}} A)$, $p \in (0,1]$, for unconditional convex measures $\mu$ and unconditional convex body $A$ in $\mathbb{R}^n$. We also prove that the (B)-conjecture for all uniform measures is equivalent to the (B)-conjecture for all log-concave measures, completing recent works by Saroglou.