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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
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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.


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

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