Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Har-Peled, Sariel; Jones, Mitchell https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-138412
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Stabbing Convex Bodies with Lines and Flats

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Abstract

We study the problem of constructing weak ε-nets where the stabbing elements are lines or k-flats instead of points. We study this problem in the simplest setting where it is still interesting - namely, the uniform measure of volume over the hypercube [0,1]^d. Specifically, a (k,ε)-net is a set of k-flats, such that any convex body in [0,1]^d of volume larger than ε is stabbed by one of these k-flats. We show that for k ≥ 1, one can construct (k,ε)-nets of size O(1/ε^{1-k/d}). We also prove that any such net must have size at least Ω(1/ε^{1-k/d}). As a concrete example, in three dimensions all ε-heavy bodies in [0,1]³ can be stabbed by Θ(1/ε^{2/3}) lines. Note, that these bounds are sublinear in 1/ε, and are thus somewhat surprising.

BibTeX - Entry

@InProceedings{harpeled_et_al:LIPIcs.SoCG.2021.42,
  author =	{Har-Peled, Sariel and Jones, Mitchell},
  title =	{{Stabbing Convex Bodies with Lines and Flats}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{42:1--42:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13841},
  URN =		{urn:nbn:de:0030-drops-138412},
  doi =		{10.4230/LIPIcs.SoCG.2021.42},
  annote =	{Keywords: Discrete geometry, combinatorics, weak \epsilon-nets, k-flats}
}

Keywords: Discrete geometry, combinatorics, weak ε-nets, k-flats
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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