License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2018.64
URN: urn:nbn:de:0030-drops-95271
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Mustafa, Nabil H. ; Ray, Saurabh

On a Problem of Danzer

LIPIcs-ESA-2018-64.pdf (0.4 MB)


Let C be a bounded convex object in R^d, and P a set of n points lying outside C. Further let c_p, c_q be two integers with 1 <= c_q <= c_p <= n - floor[d/2], such that every c_p + floor[d/2] points of P contains a subset of size c_q + floor[d/2] whose convex-hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex-hull is disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.
In particular, our general theorem implies polynomial bounds for Hadwiger-Debrunner (p, q) numbers for balls in R^d. For example, it follows from our theorem that when p > q >= (1+beta) * d/2 for beta > 0, then any set of balls satisfying the HD(p,q) property can be hit by O(q^2 p^{1+1/(beta)} log p) points. This is the first improvement over a nearly 60-year old exponential bound of roughly O(2^d).
Our results also complement the results obtained in a recent work of Keller et al. where, apart from improvements to the bound on HD(p, q) for convex sets in R^d for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

BibTeX - Entry

  author =	{Nabil H. Mustafa and Saurabh Ray},
  title =	{{On a Problem of Danzer}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{64:1--64:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Yossi Azar and Hannah Bast and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-95271},
  doi =		{10.4230/LIPIcs.ESA.2018.64},
  annote =	{Keywords: Convex polytopes, Hadwiger-Debrunner numbers, Epsilon-nets, Balls}

Keywords: Convex polytopes, Hadwiger-Debrunner numbers, Epsilon-nets, Balls
Collection: 26th Annual European Symposium on Algorithms (ESA 2018)
Issue Date: 2018
Date of publication: 14.08.2018

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