Family Of Convex Subsets Research Articles

On the complexity of the family of convex sets in ℝ d

Estimates of quantities characterizing the complexity of the family of convex subsets of the d-dimensional cube [1, n]d as n→∞ are given. The geometric properties of spaces with norm generated by the generalized majorant of partial sums are studied.

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 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.