A strengthening of the dimensional Brunn-Minkowski conjecture implies the (B)-conjecture

TL;DR

This paper demonstrates that a stronger version of the dimensional Brunn-Minkowski conjecture for certain measures implies the (B)-conjecture, providing new insights and alternative proofs for hereditarily convex measures.

Contribution

It establishes a link between a strengthened form of the dimensional Brunn-Minkowski conjecture and the (B)-conjecture, with applications to hereditarily convex measures.

Findings

A stronger form of the dimensional Brunn-Minkowski conjecture implies the (B)-conjecture.

Hereditarily convex measures satisfy the strengthened conjecture.

Provides an alternative proof for hereditarily convex measures satisfying both conjectures.

Abstract

We prove that if a sufficiently regular even log-concave measure satisfies a certain stronger form of the dimensional Brunn-Minkowski conjecture, then it also satisfies the (B)-conjecture. Furthermore, we show that hereditarily convex measures satisfy the aforementioned strengthened form, therefore providing an alternative proof of a recent result by Cordero-Erausquin and Eskenazis stating that a hereditarily convex measure satisfies both conjectures.