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Hollow polytopes of large width

Authors: Giulia Codenotti and Francisco Santos
Journal: Proc. Amer. Math. Soc. 148 (2020), 835-850
MSC (2010): Primary 52C07, 52B20; Secondary 52C17
Published electronically: August 7, 2019
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Abstract: We construct the first known hollow lattice polytopes of width larger than dimension: a hollow lattice polytope (resp., a hollow lattice simplex) of dimension $ 14$ (resp., $ 404$) and of width $ 15$ (resp., $ 408$). We also construct a hollow (nonlattice) tetrahedron of width $ 2+\sqrt 2$, and we conjecture that this is the maximum width among $ 3$-dimensional hollow convex bodies.

We show that the maximum lattice width grows (at least) additively with $ d$. In particular, the constructions above imply the existence of hollow lattice polytopes (resp., hollow simplices) of arbitrarily large dimension $ d$ and width $ \simeq 1.14 d$ (resp., $ \simeq 1.01 d$).

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Additional Information

Giulia Codenotti
Affiliation: Institut für Mathematik, Freie Universität Berlin, Germany

Francisco Santos
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Cantabria, Santander, Spain

Received by editor(s): December 17, 2018
Received by editor(s) in revised form: April 27, 2019, April 29, 2019, and May 27, 2019
Published electronically: August 7, 2019
Additional Notes: The authors were supported by the Einstein Foundation Berlin under grant EVF-2015-230 and, while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2017 semester, by the Clay Institute and the National Science Foundation (Grant No. DMS-1440140).
The work of the second author was also supported by project MTM2017-83750-P of the Spanish Ministry of Science (AEI/FEDER, UE)
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2019 American Mathematical Society