On $p$-Brunn-Minkowski inequalities for intrinsic volumes with $0\leq p<1$

TL;DR

This paper investigates the validity of the $p$-Brunn-Minkowski inequality for intrinsic volumes of convex bodies, proving it near the unit ball for $0 extless p extless 1$, but showing it fails for general convex bodies when $p$ is close to zero.

Contribution

It establishes the validity of the $p$-Brunn-Minkowski inequality for certain intrinsic volumes near the unit ball and demonstrates its failure for general convex bodies when $p$ is near zero.

Findings

Inequality holds near the unit ball for $0 extless p extless 1$.

Inequality does not hold for all convex bodies when $p$ is close to zero.

Results clarify the range of validity for $p$-Brunn-Minkowski inequalities.

Abstract

We prove the validity of the $p$-Brunn-Minkowski inequality for the intrinsic volume $V_k$, $k=2,\dots, n-1$, of convex bodies in $\mathbb{R}^n$, in a neighborhood of the unit ball, for $0\le p<1$. We also prove that this inequality does not hold true on the entire class of convex bodies of $\mathbb{R}^n$, when $p$ is sufficiently close to $0$.