Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers

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 Carathodory, Helly, and Radon numbers is true, namely, that if ^ has Carathodory 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. l Preliminary definitions. A family of subsets of a space X which is closed under intersection yields a weak type of closure, or hull, operator on the power set of X, producing concepts which may be readily applied to convexity and topology alike. Our main interest is, however, convexity theory and the abstraction of certain classical concepts from that area. (See in this regard the papers by Danzer, Grnbaum and Klee [1], Hammer We shall, therefore, introduce the following terminology: A family ^ of subsets of a set X is termed a convexity structure for X, with the pair (X, <&) being called a convexity space, whenever the following two conditions hold: (a) 0 and X belong to <g% (b) fl^" e & for each subfamily ^~c <gf. ^ is designated 2\ iff the further condition (c) {x} e T for each xeX holds, and a subfamily & of ^ is called a basis of ^ iff each member of ^ is obtainable as an intersection of members of ^. The hull operator generated by a convexity structure ^, defined in the usual manner by the relation } , Scl, 471