Steinhaus conditions for convex polyhedra

TL;DR

This paper investigates the antipodal map on convex polyhedra, proving that while most points have unique antipodes, no convex polyhedron satisfies the Steinhaus condition where the antipodal map is an involution.

Contribution

It establishes that convex polyhedra have dense sets of points with unique antipodes but do not meet the Steinhaus condition, advancing understanding of antipodal properties in convex geometry.

Findings

Most points on convex polyhedra have unique antipodes.

Convex polyhedra do not satisfy the Steinhaus condition.

The antipodal map is an involution only on a measure-zero set.

Abstract

On a convex surface $S$, the antipodal map $F$ associates to a point $p$ the set of farthest points from $p$, with respect to the intrinsic metric. $S$ is called a Steinhaus surface if $F$ is a single-valued involution. We prove that any convex polyhedron has an open and dense set of points $p$ admitting a unique antipode $F_p$, which in turn admits a unique antipode $F_{F_p}$, distinct from $p$. In particular, no convex polyhedron is Steinhaus.