When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.42
URN: urn:nbn:de:0030-drops-138412
URL: https://drops.dagstuhl.de/opus/volltexte/2021/13841/
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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},
}`