Normal polytopes and ellipsoids

TL;DR

This paper investigates the conditions under which lattice polytopes, especially those related to ellipsoids, have unimodular covers and normality, revealing new insights into their geometric and combinatorial properties.

Contribution

It establishes criteria for unimodular coverings in 3-polytopes and demonstrates the existence of non-normal lattice point convex hulls in higher-dimensional ellipsoids.

Findings

Unimodular simplices cover neighborhoods of 3-polytope boundaries if and only if the polytope is very ample.

Convex hulls of lattice points in ellipsoids in R^3 have unimodular covers.

In dimensions ≥5, there exist ellipsoids with non-normal lattice point convex hulls.

Abstract

We show that: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary of the polytope if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in R^3 has a unimodular cover, and (3) for every d at least 5, there are ellipsoids in R^d, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (3) answers a question of Bruns, Michalek, and the author.