Metric spaces in which minimal circuits cannot self-intersect
HTML articles powered by AMS MathViewer
- by David Sanders
- Proc. Amer. Math. Soc. 56 (1976), 383-387
- DOI: https://doi.org/10.1090/S0002-9939-1976-0414425-8
- PDF | Request permission
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
- Karl Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 103 (1930), no. 1, 466–501 (German). MR 1512632, DOI 10.1007/BF01455705
- L. V. Quintas and Fred Supnick, On some properties of shortest Hamiltonian circuits, Amer. Math. Monthly 72 (1965), 977–980. MR 188872, DOI 10.2307/2313333
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- 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