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

Averkov, Gennadiy; Hofscheier, Johannes; Nill, Benjamin

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. Following an argument of Eisenbrand and Shmonin, we prove that every lattice polytope contains a minimal generating set of the affine lattice spanned by its lattice points such that the number of generators (and the lattice width of their convex hull) 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.

Citation

Averkov, G., Hofscheier, J., & Nill, B. (2023). Generalized flatness constants, spanning lattice polytopes, and the Gromov width. manuscripta mathematica, 170(1-2), 147-165. https://doi.org/10.1007/s00229-021-01363-x

• Journal Article Type: Article
• Acceptance Date: Nov 18, 2021
• Online Publication Date: Dec 31, 2021
• Publication Date: 2023-01
• Deposit Date: Feb 13, 2025
• Publicly Available Date: Feb 13, 2025
• Journal: manuscripta mathematica
• Print ISSN: 0025-2611
• Electronic ISSN: 1432-1785
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 170
• Issue: 1-2
• Pages: 147-165
• DOI: https://doi.org/10.1007/s00229-021-01363-x
• Public URL: https://nottingham-repository.worktribe.com/output/45311927
• Publisher URL: https://link.springer.com/article/10.1007/s00229-021-01363-x