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

TL;DR

This paper reviews historical and recent results on the average number of normals through points of convex bodies, highlighting connections to rigid body equilibria and surface evolution PDEs.

Contribution

It provides a concise summary of existing results and introduces new developments, linking the problem to other areas in geometry and physics.

Findings

Summarizes key results on normals of convex bodies

Establishes connections to static equilibria of rigid bodies

Links to geometric PDEs of surface evolution

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.