Horizontal miniatures and normal-sized miniatures of convex lattice polytopes

Abstract

Let $n,$ $d,$ and $r$ be integers such that $0 \leq r \leq d \leq n,$ and let $P \subset \mathbb R^n$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the $r$-dimensional volume of a normal-sized miniature of $P$ to that of $P$ is given by $1:\binom{d+r+1}{r},$ which generalizes the author's previous results on the volumes of the unit hypercube and lattice simplices in the case where $r = d = n.$ 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.