Boundary restricted Brunn-Minkowski inequalities

TL;DR

This paper establishes sharp lower bounds for the volume of Minkowski sums of boundaries of convex sets, advancing understanding of boundary interactions in convex geometry.

Contribution

It introduces new volumetric lower bounds for Minkowski sums of boundaries of convex bodies, addressing a question posed by V. Milman.

Findings

Proved sharp volumetric lower bounds for boundary Minkowski sums

Extended results to convex sets with connected boundaries

Provided insights into boundary volume interactions in convex geometry

Abstract

In this paper we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V.~Milman regarding the volume of $\partial K+ \partial T$ where $K$ and $T$ are convex bodies, we prove sharp volumetric lower bounds for the Minkowski average of the boundaries of sets with connected boundary, as well as some related results.