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Proceedings of the American Mathematical Society

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Metric spaces in which minimal circuits cannot self-intersect


Author: David Sanders
Journal: Proc. Amer. Math. Soc. 56 (1976), 383-387
MSC: Primary 05C35; Secondary 54E35
DOI: https://doi.org/10.1090/S0002-9939-1976-0414425-8
MathSciNet review: 0414425
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Abstract: Definitions are given for self-intersecting polygons and cogeodesic points in terms of betweenness, and then it is proved that the metric spaces in which minimal polygons on a finite number of distinct noncogeodesic points are not self-intersecting are completely characterized as those metric spaces which have the following betweenness property for any four distinct points: if $ b$ is between $ a$ and $ c$ and between $ a$ and $ d$ then either $ c$ is between $ a$ and $ d$ or $ d$ is between $ a$ and $ c$.


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

  • [1] K. Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 103 (1930), 466-501. MR 1512632
  • [2] L. V. Quintas and F. Supnick, On some properties of shortest Hamiltonian circuits, Amer. Math. Monthly 72 (1965), 977-980. MR 32 #6304. MR 0188872 (32:6304)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0414425-8
Keywords: Non-self-intersecting polygon, noncogeodesic points, geodesic metric space
Article copyright: © Copyright 1976 American Mathematical Society

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