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Characterizing a circle with the double midset property


Authors: L. D. Loveland and J. E. Valentine
Journal: Proc. Amer. Math. Soc. 53 (1975), 443-444
MSC: Primary 52A05; Secondary 54E40
DOI: https://doi.org/10.1090/S0002-9939-1975-0388242-0
MathSciNet review: 0388242
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Abstract: A short and elementary proof is given to show that a space $ X$ is a circle with the natural geodesic metric if $ X$ is a nondegenerate, complete, convex metric space with the double midset property.


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

  • [1] A. D. Berard, Jr., Characterizations of metric spaces by the use of their midsets: One-spheres, Notices Amer. Math. Soc. 19 (1972), A-198. Abstract #691-54-11.
  • [2] -, Characterizations of metric spaces by the use of their midsets: One-spheres (Unpublished manuscript, 1-14).
  • [3] A. D. Berard, Jr. and W. Nitka, A new definition of the circle by the use of bisectors, Fund. Math. 85 (1974), 49-55. MR 0355991 (50:8464)
  • [4] L. M. Blumenthal, Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. MR 14, 1009. MR 0054981 (14:1009a)
  • [5] L. D. Loveland and J. E. Valentine, Convex metric spaces with 0-dimensional midsets, Proc. Amer. Math. Soc. 37 (1973), 568-571. MR 46 #9915. MR 0310817 (46:9915)
  • [6] L. D. Loveland and S. G. Wayment, Characterizing a curve with the double midset property, Amer. Math. Monthly 81 (1974), 1003-1006. MR 0418059 (54:6103)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0388242-0
Keywords: Convex, midsets, bisectors, simple closed curves, double midset property
Article copyright: © Copyright 1975 American Mathematical Society

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