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 | References | Similar Articles | Additional Information
Abstract: A short and elementary proof is given to show that a space 
is a circle with the natural geodesic metric if 
is a nondegenerate, complete, convex metric space with the double midset property.
- [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- [2] -, Characterizations of metric spaces by the use of their midsets: One-spheres (Unpublished manuscript, 1-14).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), no. 1, 49–55. MR 0355991
- [4] Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
- [5] L. D. Loveland and J. E. Valentine, Convex metric spaces with 0-dimensional midsets, Proc. Amer. Math. Soc. 37 (1973), 568–571. MR 0310817, https://doi.org/10.1090/S0002-9939-1973-0310817-3
- [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, https://doi.org/10.2307/2319308
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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