Average-sized miniatures and normal-sized miniatures of lattice polytopes

TL;DR

This paper investigates the ratios of volumes between lattice polytopes and their special miniatures, providing explicit ratios for squares, simplices, and hypercubes, advancing understanding of lattice polytope geometry.

Contribution

It introduces the concepts of average-sized and normal-sized miniatures of lattice polytopes and derives explicit volume ratios for these miniatures in specific cases.

Findings

For lattice squares in R^2, the volume ratio of an average-sized miniature to the original is 2:15.

For lattice simplices in R^d, the volume ratio of a normal-sized miniature to the original is 1:binomial(2d+1, d).

The ratio for hypercubes matches the known result, confirming the generality of the findings.

Abstract

Let $d \geq 0$ be an integer and let $P \subset \mathbb R^d$ be a $d$-dimensional lattice polytope. We call a polytope $M \subset \mathbb R^d$ such that $M \subset P$ and $M \sim P$ a {\itshape miniature} of $P,$ and it is said to be {\itshape horizontal} if $M$ is transformed into $P$ by translating and rescaling. A miniature $M$ of $P$ is said to be {\itshape average-sized} (resp.~{\itshape normal-sized}) if the volume of $M$ is equal to the limit of the sequence whose $n$-th term is the average of the volumes of all miniarures (resp.~all horizontal miniatures) whose vertices belong to $(n^{-1}\mathbb Z)^d.$ We prove that, for any lattice square $P \subset \mathbb R^2,$ the ratio of the areas of an average-sized miniature of $P$ and $P$ is $2:15.$ We also prove that, for any lattice simplex $P \subset \mathbb R^d,$ the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d}.$ This ratio is same as the known result for the hypercube $[0,1]^d$ provided by the author.