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

TL;DR

This paper studies mu_av and mu_nl, two asymptotic volume functionals for lattice polytopes defined via resolution-n miniatures. It proves existence and similarity-invariance properties and computes exact constants: for lattice squares, mu_av(P) = (2/15) area; for lattice simplices, mu_nl(P) = vol(P)/binom(2d+1,d). The results extend a prior hypercube case to general squares and simplices, using combinatorial counting of discretized miniatures and asymptotic analysis. The work links discrete discretizations to continuous geometry with explicit constants, relevant to lattice polytope theory and geometric combinatorics.

Abstract

Let $d \geq 0$ be an integer and $P \subset \mathbb R^d$ 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 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 $(\mathbb Z[1/n])^d.$ 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 volumes of a normal-sized miniature of $P$ and $P$ is $1:\binom{2d+1}{d}.$ This ratio is the same as the known result for the hypercube $[0,1]^d$ provided by the author.

Theorems & Definitions (25)

• Theorem 1: Hir
• Definition 1
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• Definition 5
• Proposition 1
• Theorem 2
• Theorem 3
• Lemma 1