Ball characterizations in planes and spaces of constant curvature, I

Let us have in S², R² or H² a pair of convex bodies, for S² different from S², 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 S², R² or H² 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 R², parallel strips. We describe all pairs of such subsets K, L, whose any congruent copies have an intersection with axial symmetry. For S², R² and H² there are 1, 5 and 9 cases, resp. Let us have in Sᵈ, Rᵈ or Hᵈ proper closed convex C²_+ 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 Sᵈ, Rᵈ or Hᵈ, which have at any boundary points supporting spheres, for Sᵈ of radius less than /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 Sᵈ of radius less than /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 S² of radius less than /2, and for S² smoothness of K and L. If we suppose axial symmetry of all the above convex hulls, then our bodies are circles, for S² of radii less than /2.