On the Vector Sum of Two Convex Sets in Space

Abstract

In the paper [KIS2], C. Kiselman studied the boundary smoothness of the vector sum of two smoothly bounded convex sets A and B in . He discovered the startling fact that even when A and B have real analytic boundary the set A + B need not have boundary smoothness exceeding C20/3 (this result is sharp). When A and B have C∞ boundaries, then the smoothness of the sum set breaks down at the level C5 (see [KIS2] for the various pathologies that arise).