Journal of Convex Analysis 26 (2019), No. 4, 1077--1088
Copyright Heldermann Verlag 2019
Symmetries of Convex Sets in the Hyperbolic Plane
Jesus Jerónimo-Castro
Facultad de Ingenieria, Universidad Autonoma de Querétaro, Querétaro, México
jesusjero@hotmail.com
Francisco G. Jimenez-Lopez
Facultad de Ingenieria, Universidad Autonoma de Querétaro, Querétaro, México
fjimenez@uaq.mx
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.
Keywords: Hyperbolic disc, orthogonal symmetry, hyperbolic symmetry.
MSC: 52A10, 52A55
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