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

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?)


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

American Mathematical Society