Horizontal miniatures and normal-sized miniatures of convex lattice polytopes

TL;DR

This paper proves a volume ratio for normal-sized miniatures of convex lattice polytopes, generalizing known results for hypercubes and simplices, using Ehrhart theory.

Contribution

It establishes a precise volume ratio for miniatures of convex lattice polytopes and connects the count of miniatures to Ehrhart polynomial properties.

Findings

Volume ratio of miniatures to original polytopes is 1 to binomial coefficient.

Number of horizontal miniatures with resolution t is a polynomial of degree d+1.

Leading coefficient of this polynomial relates to the volume of the polytope.

Abstract

Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$ which generalizes the known results for the unit hypercube and lattice simplices provided by the author. This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\mathrm{vol}\,(P)/(d+1),$ which is derived from Ehrhart theory.