Is there a smooth lattice polytope which does not have the integer decomposition property?

TL;DR

This paper explores Tadao Oda's longstanding question about the existence of smooth lattice polytopes lacking the integer decomposition property, connecting classical geometric results with modern lattice polytope theory.

Contribution

It introduces the problem, reviews related two-dimensional cases, and discusses the open question's significance in lattice polytope research.

Findings

Analysis of two-dimensional lattice polygons and Pick's Theorem

Discussion of the open problem's implications for higher dimensions

Clarification of the relationship between smoothness and the integer decomposition property

Abstract

We introduce Tadao Oda's famous question on lattice polytopes which was originally posed at Oberwolfach in 1997 and, although simple to state, has remained unanswered. The question is motivated by a discussion of the two-dimensional case - including a proof of Pick's Theorem, which elegantly relates the area of a lattice polygon to the number of lattice points it contains in its interior and on its boundary.