The diameter of lattice zonotopes
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- by Antoine Deza, Lionel Pournin and Noriyoshi Sukegawa PDF
- Proc. Amer. Math. Soc. 148 (2020), 3507-3516 Request permission
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
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Additional Information
- Antoine Deza
- Affiliation: McMaster University, Hamilton, Ontario, Canada
- MR Author ID: 354865
- ORCID: 0000-0002-2392-4607
- Email: deza@mcmaster.ca
- Lionel Pournin
- Affiliation: Université Paris 13, Villetaneuse, France
- MR Author ID: 810224
- Email: lionel.pournin@univ-paris13.fr
- Noriyoshi Sukegawa
- Affiliation: Tokyo University of Science, Katsushika-ku, Japan
- MR Author ID: 972693
- Email: sukegawa@rs.tus.ac.jp
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3507-3516
- MSC (2010): Primary 52B11, 11H06, 05C12
- DOI: https://doi.org/10.1090/proc/14977
- MathSciNet review: 4108856