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Bounds for Totally Separable Translative Packings in the Plane

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Abstract

A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by Fejes Tóth and Fejes Tóth (Acta Math Acad Sci Hungar 24(1–2): 229–232, 1973) and has attracted significant attention. We prove an analogue of Oler’s inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Finally, we determine the largest covering ratio (that is, the largest fraction of the plane covered by the soft disks) of an arbitrary totally separable soft disk packing with given soft parameter.

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Acknowledgements

We are indebted to the anonymous referees for careful reading and valuable comments. K. Bezdek: Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. Zs. Lángi: Partially supported by the National Research, Development and Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and grant BME FIKP-VÍZ by EMMI.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

    Károly Bezdek

  2. Department of Mathematics, University of Pannonia, Veszprém, Hungary

    Károly Bezdek

  3. MTA-BME Morphodynamics Research Group and Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary

    Zsolt Lángi

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  1. Károly Bezdek
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  2. Zsolt Lángi
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Correspondence to Károly Bezdek.

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Bezdek, K., Lángi, Z. Bounds for Totally Separable Translative Packings in the Plane. Discrete Comput Geom 63, 49–72 (2020). https://doi.org/10.1007/s00454-018-0029-6

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  • DOI: https://doi.org/10.1007/s00454-018-0029-6

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