Locating diametral points

TL;DR

This paper establishes precise conditions based on angles at boundary points of convex bodies in 2D and 3D to identify endpoints of diameters, extending to convex surfaces, and shows such criteria fail for four or more points.

Contribution

It provides sharp angle-based criteria for locating diameter endpoints on convex bodies in 2D and 3D, including convex surfaces, and proves the non-existence of such criteria for larger point sets.

Findings

Conditions based on angle sums identify diameter endpoints in 2D and 3D.

Criteria extend to convex surfaces with intrinsic metrics.

No such criteria exist for sets of four or more 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 ${\rm bd}K$, 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 $n\geq 4$ points.