J. Jerónimo-Castro, E. Makai Jr
Published 2017-08-03Version 1
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. If 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, then our sets are congruent balls.
Comments: 30 pages. arXiv admin note: text overlap with arXiv:1601.04494
Ball characterizations in planes and spaces of constant curvature, II \vskip.1cm \centerline{\rm{This pdf-file is not identical with the printed paper.}}
Ball characterizations in planes and spaces of constant curvature, I
Pairs of convex bodies in $S^d$, ${\Bbb R}^d$ and $H^d$, with symmetric intersections of their congruent copies
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