Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality

TL;DR

This paper proves the equivalence between the global and local versions of the $L^p$-Brunn-Minkowski inequality for polytopes, confirming a conjecture and providing a new proof in two dimensions.

Contribution

It establishes the equivalence of the global and local $L^p$-Brunn-Minkowski inequalities and proves the local inequality in two dimensions, settling a key conjecture.

Findings

Proved the equivalence of the global and local $L^p$-Brunn-Minkowski inequalities.

Established the local inequality in dimension 2.

Provided a new proof of the $L^p$-Brunn-Minkowski inequality in the plane.

Abstract

By studying $L^p$-combinations of strongly isomorphic polytopes, we prove the equivalence of the $L^p$-Brunn-Minkowski inequality conjectured by B\"or\"oczky, Lutwak, Yang and Zhang to the local version of the inequality studied by Colesanti, Livshyts, and Marsiglietti and by Kolesnikov and Milman, settling a conjecture of the latter authors. In addition, we prove the local inequality in dimension $2$, yielding a new proof of the $L^p$-Brunn-Minkowski inequality in the plane.