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)

 

 

A characterization of $ 2$-dimensional spherical space


Authors: L. D. Loveland and J. E. Valentine
Journal: Proc. Amer. Math. Soc. 38 (1973), 598-604
MSC: Primary 52A50
DOI: https://doi.org/10.1090/S0002-9939-1973-0320899-0
MathSciNet review: 0320899
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The midset of two distinct points $ a$ and $ b$ of a metric space is defined as the set of all points $ x$ in the space for which the distances $ ax$ and $ bx$ are equal. A metric space is said to have the $ 1$-WLMP if the midset of each two distinct points is a convex $ 1$-sphere having the property that each nonmaximal (with respect to inclusion) segment intersecting it twice lies in it. We show that a nondegenerate compact space $ X$ is isometric to a $ 2$-dimensional spherical space $ {S_{2,\alpha }}$ (a $ 2$-dimensional sphere of radius $ \alpha $ in euclidean $ 3$-space with the ``shorter arc'' metric) if and only if $ X$ has a metric with the $ 1$-WLMP.


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

  • [1] Anthony D. Berard Jr., Characterizations of metric spaces by the use of their midsets: Intervals, Fund. Math. 73 (1971/72), no. 1, 1–7. MR 0295300
  • [2] -, Characterizations of metric spaces by the use of their midsets: One spheres, Fund. Math. (to appear).
  • [3] Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
  • [4] Herbert Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR 0075623
  • [5] Casimir Kuratowski, Topologie. Vol. II, Troisième édition, corrigèe et complétée de deux appendices. Monografie Matematyczne, Tom 21, Państwowe Wydawnictwo Naukowe, Warsaw, 1961 (French). MR 0133124

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A50

Retrieve articles in all journals with MSC: 52A50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0320899-0
Article copyright: © Copyright 1973 American Mathematical Society