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 ^ 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, ^). In this connection, Levi proved that a Radon number r for ^ implies a Helly number h ^ r — 1; it is shown in this paper that exactly one additional relationship among the Carathέodory, Helly, and Radon numbers is true, namely, that if ^ has Carathέodory number c and Helly number h then ^ has Radon number r ^ ch+1. Further, characterizations of (finite) Caratheodory, 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.