A characterization of the hyperbolic disc among constant width bodies
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In this paper we prove a condition under which a hyperbolic starshaped set has a center of hyperbolic symmetry. We also give the definition of isometric diameters for a hyperbolic convex set, which behave similar to affine diameters for Euclidean convex sets. Using this concept, we give a definition of constant hyperbolic width and we prove that the only hyperbolic sets with constant hyperbolic width and with a hyperbolic center of symmetry are hyperbolic discs.
Abstract
In this paper we prove a condition under which a hyperbolic starshaped set has a center of hyperbolic symmetry. We also give the definition of isometric diameters for a hyperbolic convex set, which behave similar to affine diameters for Euclidean convex sets. Using this concept, we give a definition of constant hyperbolic width and we prove that the only hyperbolic sets with constant hyperbolic width and with a hyperbolic center of symmetry are hyperbolic discs.
참고문헌 (13)
- A. F. Beardon, "The Geometry of Discrete Groups" (1995), Springer-Verlag
- G. D. Chakerian, "Convexity and its Applications" (1983), Birkh¨auser
- B. V. Dekster, "Double normals characterize bodies of constant width in Riemannian manifolds, Geometric analysis and nonlinear partial differential equations (Denton, TX, 1990)", Lecture Notes in Pure and Appl. Math., 144, pp.187~201 (1993), Dekker
- P. C. Hammer, "Diameters of convex bodies", Proc. Amer. Math. Soc, 5, pp.304~306 (1954)
- J. Fillmore, "Barbier’s theorem in the Lobachevsky plane", Proc. Amer. Math. Soc, 24, pp.705~709 (1970)
- J. Jer´onimo-Castro, "A characteristic property of the Euclidean disc", Period. Math. Hungar, 59(2), pp.213~222 (2009)
- J. Jer´onimo-Castro, "The equal tangents prop-erty", Adv. Geom, 14(3), pp.447~453 (2014)
- K. Leichtweiss, "Curves of constant width in the non-Euclidean geometry", Abh. Math. Sem. Univ. Hamburg, 75, pp.257~284 (2005)
- H. Rademacher, "The Enjoyment of Mathematics" (1957), Princeton University Press
- V. A. Toponogov, "Differential Geometry of Curves and Surfaces, a Concise Guide" (2006), Birkh¨auser
- L. A. Santal´o, "Note on convex curves in the hyperbolic plane", Bull. Amer. Math. Soc, 51, pp.405~412 (1945)
- L. A. Santal´o, "Convexity in the hyperbolic plane", Univ. Nac. Tucum´an Rev. Ser. A, 19, pp.173~183 (1969)
- I. Yaglom, "Convex Figures" (1961), Holt Rinehart and Winston