A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that ho lds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two fin ite sets, one of which has full dimension, and, in fact, a method for compu ting the exact lower bound in this situation, given the dimension of the la ttice and the cardinalities of the two sets. These bounds in turn imply cor responding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.