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 c and Helly number h then 𝒞 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.
