Ball Characterizations in Planes and Spaces of Constant Curvature, I

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. 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 (incongruent) 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. For ℝ 2 we exactly describe all pairs of such subsets 𝐾, 𝐿, whose any congruent copies have an intersection with axial symmetry (there are five cases).