Boundary restricted Brunn-Minkowski inequalities

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 $\mathrm{\partial }K+\mathrm{\partial }T$ 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.