Convexity in discrete space

Abstract

This paper looks at Coppel's axioms for convexity, and shows how they can be applied to discrete spaces. Two structures for a discrete geometry are considered: oriented matroids, and cell complexes. Oriented matroids are shown to have a structure which naturally satisfies the axioms for being a convex geometry. Cell complexes are shown to give rise to various different notions of convexity, one of which satisfies the convexity axioms, but the others also provide valid notions of convexity in particular contexts. Finally, algorithms are investigated to validate the sets of a matroid, and to compute the convex hull of a subset of an oriented matroid. © Springer-Verlag Berlin Heidelberg 2003.