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Math-1886 · mathematics · Combinatorics

Integer decomposition of smooth polytopes

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Statement

Definitions and mathematical statement

A lattice polytope $P\subset\mathbb R^d$ is the convex hull of finitely many points of $\mathbb Z^d$. Assume it is full-dimensional. At a vertex $v$, each incident edge has a primitive outgoing integer direction: divide its displacement vector by the greatest common divisor of its integer coordinates. The polytope is smooth if exactly $d$ edges meet at every vertex and their primitive outgoing directions form a basis of the lattice $\mathbb Z^d$.

For an integer $m\ge1$, write $mP=\{mx:x\in P\}$. The integer decomposition property is the assertion that \[ mP\cap\mathbb Z^d =\underbrace{(P\cap\mathbb Z^d)+\cdots+ (P\cap\mathbb Z^d)}_{m\text{ summands}} \quad\text{for every }m\ge1. \] Does every smooth lattice polytope have this property, in every dimension?

Repeated summands are allowed. The inclusion from right to left follows from convexity; the problem is to express each lattice point of every dilate as a sum of lattice points of the original polytope. Smoothness is a local integral condition at each vertex, whereas the conclusion concerns all lattice points globally. Requiring only real summands would be automatic, and checking a few small dilates does not establish the universal quantifier over $m$ without an additional argument.

Short English statement

Does the local lattice smoothness of a polytope ensure that every lattice point in every integer dilate decomposes into lattice points of the original polytope?

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