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New covering and illumination results for a class of polytopes

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Abstract

In this paper, we focus on covering and illumination properties of a specific class of convex polytopes denoted by \(\mathcal {P}\). These polytopes are obtained as the convex hull of the Minkowski sum of a finite subset of \(\mathbb {Z}^n\) and \((1/2)[-1,1]^n\). Our investigation includes the verification of Hadwiger’s covering conjecture for \(\mathcal {P}\), as well as the estimation of the covering functional for convex polytopes in \(\mathcal {P}\). Furthermore, we demonstrate that when an integer M is sufficiently large, the elements belonging to \(\mathcal {P}\) that are contained in \(M[-1,1]^n\) serve as an \(\varepsilon \)-net for the space of convex bodies in \(\mathbb {R}^n\), equipped with the Banach–Mazur metric.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant number 12071444), the Fundamental Research Program of Shanxi Province (Grant numbers 202103021223191 and 202303021221116), and the 19th Graduate Science and Technology Project of North University of China (Grant number 20231944).

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Authors and Affiliations

  1. School of Mathematics, North University of China, Taiyuan, 030051, China

    Shenghua Gao, Senlin Wu & Longzhen Zhang

  2. Fakultüt für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Horst Martini

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  1. Shenghua Gao
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  2. Horst Martini
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  3. Senlin Wu
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  4. Longzhen Zhang
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Correspondence to Senlin Wu.

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Gao, S., Martini, H., Wu, S. et al. New covering and illumination results for a class of polytopes. Arch. Math. (2024). https://doi.org/10.1007/s00013-024-01985-z

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  • DOI: https://doi.org/10.1007/s00013-024-01985-z

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