Separation of Two Convex Sets in Convexity Structures
نویسنده
چکیده
A convexity structure satisfies the separation property 5'4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces, n-ary convexities, and graphs. In particular, it is proven that a) an n-ary convexity is $4 iff every pair of disjoint polytopes with at most n vertices can be separated by complementary half spaces, and b) an interval convexity is $4 iff it satisfies the analogue of the Pasch axiom of plane geometry: A characterization of bipartite and weakly modular spaces with S4 convexity is given in terms of forbidden subgraphs. 1INTRODUCTION Separation of convex sets constitutes one of the fundamental facets of the theory of convex sets in linear spaces. In order to obtain analogous results for other types of convexities notions such as hyperplane and half space have to be defined suitably. For convexities such as convexity spaces [1,9,15,17,28], join spaces [28], metric convexity of finite-dimensional normed spaces [8,31] various separation theorems by hyperplanes were obtained~ For general convexity structures, discrete, topological or metric convexities, separation is usuMy The purpose of this paper is to present criteria of separation by half spaces in various classes of convexity structures. The work has gone on for several years and this paper will serve to present the obtained results. Some of the results have appeared before in the Moldow State University press [10,11,14], others previously have stated without proof.
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