Ball characterizations in planes and spaces of constant curvature, I

TL;DR

This paper characterizes convex bodies in constant curvature spaces based on symmetry properties of their intersections and convex hulls, identifying conditions under which bodies are congruent circles or spheres.

Contribution

It provides a comprehensive classification of convex bodies in spherical, Euclidean, and hyperbolic spaces using symmetry of intersections and convex hulls, extending classical geometric results.

Findings

Convex bodies with centrally symmetric intersections are congruent circles in S^2, R^2, or H^2.

Bodies with axially symmetric intersections are circles or parallel strips.

Small intersection symmetry implies boundary components are spheres, paraspheres, or hyperspheres.

Abstract

Let us have in S^2, R^2 or H^2 a pair of convex bodies, for S^2 different from S^2, such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any congruent copies of them are axially symmetric, then our bodies are circles. Let us have in S^2, R^2 or 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. If the intersections of any congruent copies of K and L are centrally symmetric, then K and L are congruent circles, or, for R^2, parallel strips. We describe all pairs of such subsets K,L, whose any congruent copies have an intersection with axial symmetry. For S^2, R^2 and H^2 there are 1, 5 and 9 cases, resp. Let us have in S^d, 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, 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 circles, for S^2 of radii less than \pi/2.