Ehrhart-equivalent $\boldsymbol 3$-polytopes are equidecomposable

TL;DR

This paper proves that in three dimensions, lattice polytopes with identical Ehrhart functions can be partitioned into equivalent unimodular simplices, establishing a strong geometric equivalence based on Ehrhart data.

Contribution

It demonstrates that Ehrhart-equivalent 3-polytopes are necessarily $ ext{GL}_3( ext{Z})$-equidecomposable, linking Ehrhart theory with geometric decompositions.

Findings

Ehrhart-equivalent 3-polytopes are equidecomposable.

Partition into unimodular simplices preserves Ehrhart equivalence.

Provides a geometric criterion for Ehrhart equivalence in 3D.

Abstract

We show that if two lattice $3$-polytopes $P$ and $P'$ have the same Ehrhart function then they are $\operatorname{GL}_3({\mathbb Z})$-equidecomposable; that is, they can be partitioned into relatively open simplices $U_1,\dots, U_k$ and $U'_1,\dots,U'_k$ such that $U_i$ and $U'_i$ are unimodularly equivalent, for each $i$.