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 $\partial K+ \partial T$ 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.
Comment: Previous version extended and split into three papers. This is paper no. 1; 13 pages, 3 figures, published in Communications in Contemporary Mathematics