Smooth centrally symmetric polytopes in dimension 3 are IDP

TL;DR

This paper proves Oda's conjecture for smooth, centrally symmetric 3D lattice polytopes, demonstrating they possess the integer decomposition property by covering them with lattice parallelepipeds and unimodular simplices.

Contribution

It establishes the conjecture for a specific class of 3D polytopes, advancing understanding of the integer decomposition property in lattice polytopes.

Findings

Centrally symmetric smooth 3D polytopes are covered by lattice parallelepipeds and unimodular simplices.

These polytopes satisfy the integer decomposition property.

The proof confirms Oda's conjecture in this specific case.

Abstract

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.