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

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.