Geodesic
Symmetries of Convex Sets in the Hyperbolic Plane
Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1077-1088
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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.
Classification : 52A10, 52A55
Mots-clés : Hyperbolic disc, orthogonal symmetry, hyperbolic symmetry 
@article{JCA_2019_26_4_JCA_2019_26_4_a3,
     author = {J. Jer\'onimo-Castro and F. G. Jimenez-Lopez},
     title = {Symmetries of {Convex} {Sets} in the {Hyperbolic} {Plane}},
     journal = {Journal of convex analysis},
     pages = {1077--1088},
     year = {2019},
     volume = {26},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a3/}
}
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J. Jerónimo-Castro; F. G. Jimenez-Lopez. Symmetries of Convex Sets in the Hyperbolic Plane. Journal of convex analysis, Tome 26 (2019) no. 4, pp. 1077-1088. http://geodesic.mathdoc.fr/item/JCA_2019_26_4_JCA_2019_26_4_a3/