Ball characterizations in planes and spaces of constant curvature, II \vskip.1cm \centerline{\rm{This pdf-file is not identical with the printed paper.}}

TL;DR

This paper extends High's theorem on convex bodies and symmetry from the plane to spaces of constant curvature, characterizing convex sets with symmetric intersections and hulls in spherical, Euclidean, and hyperbolic geometries.

Contribution

It generalizes symmetry theorems for convex bodies to curved spaces and classifies pairs of convex sets with symmetric intersections and hulls in these geometries.

Findings

Characterization of convex sets with symmetric intersections in hyperbolic space.

Classification of pairs of convex bodies with symmetric convex hulls in spherical and hyperbolic spaces.

Conditions under which convex bodies are congruent spheres or balls based on symmetry 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 the sphere and the hyperbolic plane, and partly to spaces of constant curvature. We also investigate the dual question about the convex hull of the unions, rather than the intersections. Let us have in $H^2$ proper closed convex subsets $K,L$ with interior points, such that the numbers of the connected components of the boundaries of $K$ and $L$ are finite. We exactly describe all pairs of such subsets $K,L$, whose any congruent copies have an intersection with axial symmetry; there are nine cases. (The cases of $S^2$ and ${\Bbb{R}}^2$ were described in Part I, i.e., \cite{5}.) Let us have in $S^d$, ${\Bbb{R}}^d$ or $H^d$ proper closed convex $C^2_+$ subsets $K,L$ with interior points, such that all sufficiently small intersections of their congruent copies are symmetric w.r.t.\ a particular hyperplane. Then the boundary components of both $K$ and $L$ are congruent, and each of them is a sphere, a parasphere or a hypersphere. Let us have a pair of convex bodies in $S^d$, ${\Bbb{R}}^d$ or $H^d$, which have at any boundary points supporting spheres (for $S^d$ of radius less than $\pi /2$). If the convex hull of the union of any congruent copies of these bodies is centrally symmetric, then our bodies are congruent balls (for $S^d$ of radius less than $\pi /2$). An analogous statement holds for symmetry w.r.t.\ a particular hyperplane. For $d=2$, suppose the existence of the above supporting circles (for $S^2$ of radius less than $\pi /2$), and, for $S^2$, smoothness of $K$ and $L$. If we suppose axial symmetry of all the above convex hulls, then our bodies are (incongruent) circles (for $S^2$ of radii less than $\pi /2$).