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.
Highlights
A lattice polytope in Rd is the convex hull of finitely many points in the integer lattice Zd
All polytopes in this paper will be assumed to be lattice polytopes. They appear naturally in a variety of different fields, such as combinatorics, commutative algebra, toric geometry and optimization, where their geometric and arithmetic behavior has been intensively studied in recent decades
IDP polytopes are of great interest when studying the arithmetic behavior of dilated polytopes (Ehrhart theory) as well as in commutative algebra and toric geometry
Summary
A lattice polytope in Rd is the convex hull of finitely many points in the integer lattice Zd. All polytopes in this paper will be assumed to be lattice polytopes They appear naturally in a variety of different fields, such as combinatorics, commutative algebra, toric geometry and optimization, where their geometric and arithmetic behavior has been intensively studied in recent decades. Smooth lattice polytopes, integer decomposition property, Oda’s conjecture, central symmetry, 3-dimensional polytopes. Due to its relation with projective normality of projective toric varieties, the following specialization of Problem 1.1 was asked by Oda [5]. It has since become known as Oda’s Conjecture.
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