  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.
