Topes of Oriented Matroids and Related Structures

Abstract

An oriented matroid can be viewed as a combinatorial abstraction of the facial incidence relations of the polyhedral cones induced by a finite arrangement of oriented hyperplanes in ${\mathrm{R}}^d$ through the origin. “Topes” of an oriented matroid correspond to maximal polyhedral cones. This paper discusses three structures related to topes of oriented matroids, namely, acycloids, $L^1$-systems and median systems. It is shown that $L^1$-systems are closely related to convex geometries. Median systems are introduced as an equivalent notion of median graphs, and they are, in particular, applied to characterize median graphs. Perturbations of acycloids and $L^1$-systems are studied.