[Paper Review] On Arnold's Problem on the Classifications of Convex Lattice Polytopes

Chuanming Zong|arXiv (Cornell University)|Jul 14, 2011

Computational Geometry and Mesh Generation10 references3 citations

TL;DR

This paper establishes a tight lower bound for the number of non-equivalent d-dimensional centrally symmetric convex lattice polytopes with a given number of lattice points, using unimodular group actions and geometric constructions. It proves that the logarithm of this count grows as $ w^{(d-1)/(d+1)} $, matching known upper bounds and resolving a long-standing conjecture for the centrally symmetric case.

ABSTRACT

In 1980, V.I. Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its analogues have been studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers of non-equivalent ddimensional convex lattice polytopes of given volume or cardinality have been achieved. In this paper, by introducing and studying the unimodular groups acting on convex lattice polytopes, we obtain lower bounds for the number of non-equivalent d-dimensional convex lattice polytopes of bounded volume or given cardinality, which are essentially tight.

Motivation & Objective

• To resolve the classification problem for d-dimensional centrally symmetric convex lattice polytopes by establishing a tight lower bound on their number.
• To extend Arnold's work on convex lattice polygons to higher dimensions and centrally symmetric polytopes.
• To prove that the logarithmic growth rate of the number of such polytope classes matches the known upper bounds, confirming tightness.
• To introduce and analyze unimodular groups acting on lattice polytopes to control equivalence classes.

Proposed method

• Introduces unimodular groups acting on convex lattice polytopes and estimates their orders to control equivalence under affine transformations.
• Constructs a family of centrally symmetric convex lattice polytopes by modifying a ball-like lattice set $ B_{d,r} \cap \mathbb{Z}^d $ and its reflection.
• Uses a geometric construction involving $ P_{d,r} $, a subset of lattice points in a ball of radius $ r $, and removes $ j $ points to form $ P'_{d,r} $.
• Applies a symmetric doubling construction: $ P_{\mathbf{v}_1,\dots,\mathbf{v}_j} = P' \cup (-P') $, ensuring central symmetry about the origin.
• Estimates the size of the family $ \mathcal{F} $ of such polytopes as $ 2^{|V'_{d,r}|} $, where $ V'_{d,r} $ is a set of lattice points in a shell of radius $ r $.
• Uses group-theoretic arguments to bound the number of equivalent polytopes under unimodular transformations, leading to the final lower bound via $ \kappa^*(d,w) \gg 2^{|V'_{d,r}|} / (2^d \cdot (d-1)!) $.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of non-equivalent d-dimensional centrally symmetric convex lattice polytopes with a fixed number of lattice points?
• RQ2 Can a lower bound matching the known upper bound of $ w^{(d-1)/(d+1)} $ be established for this count?
• RQ3 How do unimodular group actions affect the classification of lattice polytopes under affine equivalence?
• RQ4 Is the logarithmic growth rate of $ \kappa^*(d,w) $ essentially tight, as conjectured by Arnold and later extended by Bárány and Vershik?

Key findings

• The logarithm of the number of non-equivalent d-dimensional centrally symmetric convex lattice polytopes with $ w $ lattice points satisfies $ \log \kappa^*(d,w) \gg w^{(d-1)/(d+1)} $.
• This lower bound matches the previously known upper bound $ \log \kappa^*(d,w) \ll w^{(d-1)/(d+1)} $, establishing that $ \log \kappa^*(d,w) \asymp w^{(d-1)/(d+1)} $.
• The construction yields $ 2^{|V'_{d,r}|} $ distinct polytopes, with $ |V'_{d,r}| \gg r^{d-1} $, and $ r \asymp w^{1/(d+1)} $, leading to the final growth rate.
• The proof relies on controlling unimodular symmetries: only $ 2^d \cdot (d-1)! $ transformations preserve the constructed polytopes, enabling a precise count of inequivalent classes.
• The result confirms that the growth rate $ w^{(d-1)/(d+1)} $ is tight for centrally symmetric polytopes, resolving a key open problem in lattice polytope classification.
• The method also implies $ \log \left( \sum_{j=1}^m v(d,j) \right) \gg m^{(d-1)/(d+1)} $, consistent with Bárány’s earlier result but derived via a new geometric approach.