Pairs of convex bodies in $S^d$, ${\Bbb R}^d$ and $H^d$, with symmetric intersections of their congruent copies

TL;DR

This paper extends High's theorem on convex bodies with centrally symmetric intersections from the plane to spherical and hyperbolic geometries, characterizing when such bodies are congruent circles or balls.

Contribution

It generalizes a classical result to non-Euclidean geometries and provides a detailed classification of convex sets with symmetric intersections in these spaces.

Findings

In spherical, Euclidean, and hyperbolic spaces, sets with centrally symmetric intersections are congruent balls.

In 2D, the paper classifies pairs of convex sets with symmetric intersections based on boundary components.

The results include characterizations for sets with finitely many boundary components and symmetric intersection properties.

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 and hyperbolic planes. If in any of these planes, or in ${\Bbb R}^2$, there is a pair of closed convex sets with interior points, and the intersections of any congruent copies of these sets are centrally symmetric, then, under some mild hypotheses, our sets are congruent circles, or, for ${\Bbb R}^2$, two parallel strips. We prove the analogue of this statement, for $S^d$, ${\Bbb R}^d$, $H^d$, if we suppose $C^2_+$: again, our sets are congruent balls. In $S^2$, ${\Bbb R}^2$ and $H^2$ we investigate a variant of this question: supposing that the numbers of connected components of the boundaries of both sets are finite, we exactly describe all pairs of such closed convex sets, with interior points, whose any congruent copies have an intersection with axial symmetry (there are 1, 5 or 9 cases, respectively).