Generalized flatness constants, spanning lattice polytopes, and the Gromov width

TL;DR

This paper explores the properties of lattice widths in convex bodies and polytopes, establishing bounds and relations to symplectic geometry, particularly the Gromov width, and introduces the concept of generalized flatness constants.

Contribution

It introduces the study of generalized flatness constants and links lattice width properties to symplectic geometry, expanding understanding of lattice polytopes and their geometric invariants.

Findings

Convex bodies of large width contain unimodular copies of standard simplices.

Every lattice polytope has a minimal generating set with size bounded by a dimension-dependent constant.

Lattice width of Delzant polytopes relates to bounds on the Gromov width of symplectic toric manifolds.

Abstract

In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators is bounded by a constant which only depends on the dimension. We also discuss relations to recent results on spanning lattice polytopes and how our results could be viewed as the beginning of the study of generalized flatness constants. Regarding symplectic geometry, we point out how the lattice width of a Delzant polytope is related to upper and lower bounds on the Gromov width of its associated symplectic toric manifold. Throughout, we include several open questions.