Normally located polyhedra

TL;DR

This paper investigates conditions under which lattice polyhedra are normally located, establishing a link with their normal fans and providing examples where scaled triangles are not normally located.

Contribution

It proves that if the normal fan of one polyhedron refines another, then scaled versions are normally located, connecting geometric and algebraic properties.

Findings

Normal fan refinement implies scaled polyhedra are normally located.

Counterexample with scaled triangles shows not all scaled pairs are normally located.

Results connect lattice polyhedra geometry with graded algebra properties.

Abstract

Lattice polyhedra $Q_1$ and $Q_2$ with the same tail cone are said to be normally located if every lattice point in the Minkowski sum $Q_1+Q_2$ is the sum of lattice points from $Q_1$ and $Q_2$, respectively. We prove that if the normal fan of $Q_1$ refines the normal fan of $Q_2$, then there is a positive integer $k$ such that for any positive integer $s$ the polyhedra $skQ_1$ and $skQ_2$ are normally located. This result is based on an interpretation of the problem in terms of graded algebras and earlier results on surjectivity of the multiplicaiton map on homogeneous components. Also we provide an example of two lattice triangles $P$ and $Q$ on the plane such that for any positive integer $k$ the triangles $kP$ and $kQ$ are not normally located.