Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The diameter of lattice zonotopes


Authors: Antoine Deza, Lionel Pournin and Noriyoshi Sukegawa
Journal: Proc. Amer. Math. Soc. 148 (2020), 3507-3516
MSC (2010): Primary 52B11, 11H06, 05C12
DOI: https://doi.org/10.1090/proc/14977
Published electronically: March 30, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed and the number of their generators grows large. We also prove that, for infinitely many integers $ k$, the largest possible diameter $ \delta _z(d,k)$ of a lattice zonotope contained in the hypercube $ [0,k]^d$ is uniquely achieved by a primitive zonotope. We obtain, as a consequence, that $ \delta _z(d,k)$ grows like $ k^{d/(d+1)}$ up to an explicit multiplicative constant, when $ d$ is fixed and $ k$ goes to infinity, providing a new lower bound on the largest possible diameter of a lattice polytope contained in $ [0,k]^d$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52B11, 11H06, 05C12

Retrieve articles in all journals with MSC (2010): 52B11, 11H06, 05C12


Additional Information

Antoine Deza
Affiliation: McMaster University, Hamilton, Ontario, Canada
Email: deza@mcmaster.ca

Lionel Pournin
Affiliation: Université Paris 13, Villetaneuse, France
Email: lionel.pournin@univ-paris13.fr

Noriyoshi Sukegawa
Affiliation: Tokyo University of Science, Katsushika-ku, Japan
Email: sukegawa@rs.tus.ac.jp

DOI: https://doi.org/10.1090/proc/14977
Received by editor(s): June 14, 2019
Received by editor(s) in revised form: December 11, 2019
Published electronically: March 30, 2020
Additional Notes: The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant program (RGPIN-2015-06163).
The second author was partially supported by the ANR project SoS (Structures on Surfaces), grant number ANR-17-CE40-0033 and by the PHC project number 42703TD
The third author was partially supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Science Research (A) 26242027.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2020 American Mathematical Society