The Legendre Transform of Convex Lattice Sets

TL;DR

This paper explores the properties of convex lattice sets using a discrete Legendre transform, introducing a polar concept with self-duality and analyzing the Mahler product in relation to cross-polytopes.

Contribution

It defines the polar of convex lattice sets, establishes their properties, and links the Mahler product to the characterization of cross-polytopes in a discrete setting.

Findings

Polar of convex lattice sets has self-duality property.

Inclusion, union, and intersection relations are established for these polars.

The minimal Mahler product characterizes cross-polytopes.

Abstract

The goal of this paper is to study convex lattice sets by the discrete Legendre transform. The definition of the polar of convex lattice sets in $\mathbb{Z}^n$ is provided. It is worth mentioning that the polar of convex lattice sets have the self-dual property similar to that of convex bodies. Some properties of convex lattice sets are established, for instance, the inclusion relation, the union and intersection on the polar of convex lattice sets. In addition, we discuss the relationship between the cross-polytope and the discrete Mahler product. It states that a convex lattice set is the cross-polytope if and only if its discrete Mahler product is the smallest.