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

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).