Properties of Convex Lattice Sets under the Discrete Legendre Transform

Abstract

The discrete Legendre transform is a powerful tool for analyzing the properties of convex lattice sets. In this paper, for t>0, we study a class of convex lattice sets and establish a relationship between vertices of the polar of convex lattice sets and vertices of the polar of its t−dilation. Subsequently, we show that there exists a class of convex lattice sets such that its polar is itself. In addition, we calculate upper and lower bounds for the discrete Mahler product of a class of convex lattice sets.

Keywords

• convex lattice sets
• polar
• discrete Legendre transform
• discrete Mahler product