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The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent...

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Detalles Bibliográficos
Autores principales: Codenotti, Giulia, Santos, Francisco, Schymura, Matthias
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8709830/
https://www.ncbi.nlm.nih.gov/pubmed/35023883
http://dx.doi.org/10.1007/s00454-021-00330-3
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author Codenotti, Giulia
Santos, Francisco
Schymura, Matthias
author_facet Codenotti, Giulia
Santos, Francisco
Schymura, Matthias
author_sort Codenotti, Giulia
collection PubMed
description We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every [Formula: see text] . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.
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spelling pubmed-87098302022-01-10 The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices Codenotti, Giulia Santos, Francisco Schymura, Matthias Discrete Comput Geom Article We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every [Formula: see text] . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension. Springer US 2021-11-17 2022 /pmc/articles/PMC8709830/ /pubmed/35023883 http://dx.doi.org/10.1007/s00454-021-00330-3 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Codenotti, Giulia
Santos, Francisco
Schymura, Matthias
The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title_full The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title_fullStr The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title_full_unstemmed The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title_short The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices
title_sort covering radius and a discrete surface area for non-hollow simplices
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8709830/
https://www.ncbi.nlm.nih.gov/pubmed/35023883
http://dx.doi.org/10.1007/s00454-021-00330-3
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