Carathéodory, Helly, and Radon Numbers for Sublattice and Related Convexities

Abstract

The Carathéodory, Helly, and Radon numbers are three main invariants in convexity theory. These invariants have been determined, exactly or approximately, for a number of different convexity structures. We consider convexity structures defined by the sublattices and by the convex sublattices of finite-dimensional Euclidian, integer, and Boolean spaces. Such sublattices arise in submodular optimization (lattice programming) and in monotone comparative statics of optimization and fixed point problems. We also consider integral L-natural convexities, induced by dual network flow constraint systems. We determine the exact Carathéodory, Helly, and Radon numbers of most of these convexities, and very close upper and lower bounds for the other Carathéodory numbers. Our results imply, for example, that if a set can be obtained with unions and intersections from a given family of subsets of a finite set then it can be obtained with unions and intersections from a small subfamily. We also show that finding the Carathéodory number of integral L-natural convexities reduces to an extremal problem in the theory of permutations, solved in a companion paper. We leave as open problems the determination of the Helly and Radon numbers of the integer convex sublattice convexity.