  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.
