Symmetries of Convex Sets in the Hyperbolic Plane

Abstract

We establish some results characterizing central or axial symmetry of convex sets in the hyperbolic plane. The characterizations follow the spirit of a Chakerian-Klamkin's characterization of central symmetry for Euclidean sets: if for any three point-subset M of a compact set K there is a symmetric image of M that is also contained in K, then K has a center of hyperbolic symmetry. We also study axial symmetry when the axis is either a geodesic, a horocycle, or a hypercycle. Finally, in the last section we give a characterization of the hyperbolic disc.