Locating diametral points

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^d$, with $d=2,3$. We determine sharp sufficient conditions for a set $E$ composed of $1$, $2$, or $3$ points of $\mathrm{bd}K$, to contain at least one endpoint of a diameter of $K$ (for $d=2,3$). We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in $E$. We also show that such criteria do not exist for $n\ge 4$ points.

12 pages