Ball characterizations in spaces of constant curvature

TL;DR

This paper extends a classical geometric theorem to spherical, Euclidean, and hyperbolic spaces, characterizing convex sets with centrally symmetric intersections as congruent balls or related shapes under various symmetry conditions.

Contribution

It generalizes High's theorem to spaces of constant curvature, providing new characterizations of convex sets based on intersection symmetries with regularity assumptions.

Findings

Sets with centrally symmetric intersections are congruent balls in these spaces.

Under weaker symmetry conditions, sets are either balls, paraballs, or hypersphere components.

Various geometric configurations are classified, including hypercycles and parallel domains.

Abstract

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class $C^2_+$ with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either $d \ge 3$, or $d=2$ and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points --- for $S^d$ of radius less than $ \pi /2$ --- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.