Axiomatic convexity theory and relationships between the carathéodory, helly, and radon numbers

Abstract

An axiomatic setting for the theory of convexity is provided by taking an arbitrary set X and constructing a family C of subsets of X which is closed under intersections. The pair consisting of any ordered vector space and its family of convex subsets thus become the prototype for all such pairs (.X, C). In this connection, Levi proved that a Radon number r for C implies a Helly number h ≦ r — 1; it is shown in this paper that exactly one additional relationship among the Carathéodory, Helly, and Radon numbers is true, namely, that if C has Carathéodory number o and Helly number h then C has Radon number r ≦ ch+1. Further, characterizations of (finite) Carathéodory, Helly, and Radon numbers are obtained in terms of separation properties, from which emerges a new proof of Levi’s theorem, and finally, axiomatic foundations for convexity in euclidean space are discussed, resulting in a theorem of the type proved by Dvoretzky. © 1971 Pacific Journal of Mathematics.