License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.42
URN: urn:nbn:de:0030-drops-138412
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13841/
Har-Peled, Sariel ;
Jones, Mitchell
Stabbing Convex Bodies with Lines and Flats
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}
}
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Keywords: |
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Discrete geometry, combinatorics, weak ε-nets, k-flats |
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Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.06.2021 |
