Metric spaces in which minimal circuits cannot self-intersect

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 $c$.