On the classification of lattice polytopes via affine equivalence

TL;DR

This paper introduces a new affine equivalence-based framework for classifying convex lattice polytopes, providing refined results that enhance understanding of their structural classification beyond previous methods.

Contribution

The paper proposes a novel affine equivalence approach that extends existing classification results and offers deeper insights into the structure of convex lattice polytopes.

Findings

New classification results based on affine equivalence

Extension of Bárány's volume-related work

Complementary insights to Zong's cardinality studies

Abstract

In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of a given area. Since then, this problem and its analogues have been studied by many authors, including B\'ar\'any, Lagarias, Pach, Santos, Ziegler and Zong. Despite extensive study, the structure of the representative sets in the classifications remains unclear, indicating a need for refined classification methods. In this paper, we propose a novel classification framework based on affine equivalence, which offers a fresh perspective on the problem. Our approach yields several classification results that extend and complement B\'ar\'any's work on volume and Zong's work on cardinality. These new results provide a more nuanced understanding of the structure of the representative set, offering deeper insights into the classification problem.