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Proceedings of the American Mathematical Society

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



Convex metric spaces with 0-dimensional midsets

Authors: L. D. Loveland and J. E. Valentine
Journal: Proc. Amer. Math. Soc. 37 (1973), 568-571
MSC: Primary 53C70; Secondary 52A50
MathSciNet review: 0310817
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Abstract: Let X be a nontrivial, complete, convex, locally externally convex metric space. Assuming that the midset of each pair of points of X is 0-dimensional and that any nonmaximal metric segment that intersects a midset twice lies in that midset, we show that X is isometric to either the euclidean line $ {E^1}$ or to a 1-dimensional spherical space $ {S_{1,\alpha }}$ (the circle of radius $ \alpha $ in the euclidean plane with the ``shorter arc'' metric).

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Keywords: Convex, locally externally convex, midsets, weak linear midset property, 0-dimensional midsets
Article copyright: © Copyright 1973 American Mathematical Society