Abstract
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.
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