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

Abstract

Let $d\ge 0$ 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 miniature of $P,$ and it is said to be horizontal if $M$ is transformed into $P$ by translating and positive integral rescaling. A miniature $M$ of $P$ is said to be average-sized (resp. 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 $1:\left(\genfrac{}{}{0ex}{}{2d+1}{d}\right).$ This ratio is same as the known result for the hypercube $[0,1{]}^d$ provided by the author.

7 pages, 3 figures; Corrected misprints