A Brunn-Minkowski inequality for the integer lattice (Q2723461)

The main result of this paper is the following theorem which is analogous to the classical Brunn-Minkowski inequality. Let \(A\) and \(B\) be finite subsets of \(\mathbb{Z}^{n}\) with \(\dim B=n\), and denote the cardinality of a set \(X\) by \(|X|\). Then \(|A+B|\geq |D_{|A|}^{B}+D_{|B|}^{B}|\), where \(D_{|A|}^{B}\) and \(D_{|B|}^{B}\) are finite subsets of \(\mathbb{Z}^{n}\) with cardinalities equal to those of \(A\) and \(B\), respectively, that are initial segments in a certain order on \(\mathbb{Z}^{n}\) which depends only on \(|B|\). Roughly speaking, these sets are as close as possible to being the intersection with \(\mathbb{Z}^{n}\) of simplices of a certain fixed shape. NEWLINENEWLINENEWLINEThis result is applied to obtain new lower bounds for the cardinality of the Minkowski sum of two finite sets, one of which has full dimension. For example, one of these bounds reads as follows: \(|A+B|^{1/n}\geq |A|^{1/n}+((|B|-n)/n!)^{1/n}\). In fact, a method is described for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply new bounds for the lattice point enumerator of the sum of two convex lattice polytopes. NEWLINENEWLINENEWLINEIn the plane, some additional results are found, for instance the following Rogers-Shephard type inequality for the difference body \(DP\) of a convex lattice polygon: \(G(DP)\leq 6G(P)-2b(P)-5\) with equality if and only if \(P\) is a triangle. (\(G\) is the lattice point enumerator, and \(b(P)\) denotes the number of lattice points on the boundary of \(P\)).