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

Abstract: 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 n?d. 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

 Autoría: Codenotti G., Santos F., Schymura M.,

 Fuente: Discrete and Computational Geometry, 2022, 67, 65-111

 Editorial: Springer New York LLC

 Año de publicación: 2022

 Nº de páginas: 47

 Tipo de publicación: Artículo de Revista

 DOI: 10.1007/s00454-021-00330-3

 ISSN: 0179-5376,1432-0444

 Proyecto español: MTM2017-83750-P

 Url de la publicación: https://doi.org/10.1007/s00454-021-00330-3