On the classification of lattice polytopes via affine equivalence

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.