Convex normality of rational polytopes with long edges

TL;DR

This paper introduces the concept of convex normality for rational polytopes, providing bounds on edge lengths that guarantee normality, and answers a longstanding question in lattice polytope theory.

Contribution

It establishes a uniform lower bound on edge lengths ensuring convex normality and normality of lattice polytopes, advancing understanding of polytope properties.

Findings

Polytopes with edges of length at least 4d(d+1) are normal.

Lattice simplices with edges of length at least d(d+1) are covered by lattice parallelepipeds.

The approach involves rational polytopes even for lattice polytope applications.

Abstract

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope P has lattice length at least 4d(d+1) then P is normal. This answers in the positive a question raised in 2007. If P is a lattice simplex whose edges have lattice lengths at least d(d+1) then P is even covered by lattice parallelepipeds. For the approach developed here, it is necessary to involve rational polytopes even for applications to lattice polytopes.