Families of lattice polytopes of mixed degree one

Abstract

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n ≥ 4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex ∆ n−1 . Furthermore, we give a complete solution in dimension n = 3. In the course of this we show that our finiteness result does not extend to dimension n = 3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle ∆ 2 .

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

“…In the last decade, there has been an increased interest in the algorithmic theory of lattice polytopes, which is motivated by applications in algebra, algebraic geometry, combinatorics, and optimization (see, for example, [3,4,6,[8][9][10][11][12][19][20][21]23]). So far, a special emphasis has been put on computer-assisted enumeration results, which are important from different perspectives.…”