Discrete convexity and unimodularity - I

In this paper we develop a theory of convexity for the lattice of integer points Z(n), which we call theory of discrete convexity. Namely, we characterize classes of subsets of Z(n), which possess the separation property, or, equivalently, classes of integer polyhedra such that intersection of any two polyhedra of a class is an integer polyhedron (need not be in the class). Specifically, we show that these (maximal) classes are in one-to-one correspondence with pure systems. Unimodular systems constitute an important instance of pure systems. Given a unimodular system, we construct a pair of (dual) discretely convex classes, one of which is stable under summation and the other is stable under intersection. (C) 2003 Elsevier Inc. All rights reserved.