Boundary restricted Brunn-Minkowski inequalities

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.

Previous version extended and split into three papers. This is paper no. 1; 13 pages, 3 figures, published in Communications in Contemporary Mathematics