Convex metric spaces with $0$-dimensional midsets
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- by L. D. Loveland and J. E. Valentine PDF
- Proc. Amer. Math. Soc. 37 (1973), 568-571 Request permission
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).References
- 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 295300, DOI 10.4064/fm-73-1-1-7 —, Characterizations of metric spaces by the use of their midsets: One-spheres, Fund. Math. (to appear).
- Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
- Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 568-571
- MSC: Primary 53C70; Secondary 52A50
- DOI: https://doi.org/10.1090/S0002-9939-1973-0310817-3
- MathSciNet review: 0310817