Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Barbier's theorem in the Lobachevski plane

Author: Jay P. Fillmore
Journal: Proc. Amer. Math. Soc. 24 (1970), 705-709
MSC: Primary 52.25; Secondary 53.00
MathSciNet review: 0253150
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the Lobachevski plane, horocycles with the same center are geodesic parallels and are natural replacements for the lines used in defining the support function of a convex curve and the notion of constant width in the Euclidean plane. In this paper, analogs based on horocycles are obtained for Christoffel's formula, which expresses the radius of curvature of a convex curve in terms of its support function, and Barbier's theorem, which relates the length and width of a convex curve of constant width.

References [Enhancements On Off] (What's this?)

  • [1] W. J. Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika 14 (1967), 1-13. MR 36 #788. MR 0217699 (36:788)
  • [2] D. Laugwitz, Differential and Riemannian geometry, Academic Press, New York, 1965. MR 30 #2406. MR 0172184 (30:2406)
  • [3] L. A. Santaló, Note on convex curves on the hyperbolic plane, Bull. Amer. Math. Soc. 51 (1945), 405-412. MR 7, 26. MR 0012456 (7:26a)
  • [4] -, Horocycles and convex sets in hyperbolic plane, Arch. Math. (Basel) 18 (1967), 529-533. MR 37 #870. MR 0225276 (37:870)
  • [5] P. A. Širokov, A sketch of the fundamentals of Lobachevskian geometry, Noordhoff, Groningen, 1964. MR 28 #4419.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52.25, 53.00

Retrieve articles in all journals with MSC: 52.25, 53.00

Additional Information

Keywords: Convex curves, constant width, support function
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society