On the average number of normals through points of a convex body

Abstract

In 1944, Santal\'o asked about the average number of normals through a point of a given convex body. Since then, numerous results appeared in the literature about this problem. The aim of this paper is to give a concise summary of these results, with some new, recent developments. We point out connections of this problem to static equilibria of rigid bodies as well as to geometric partial differential equations of surface evolution.