On a problem of danzer

Nabil H. Mustafa, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Let C be a bounded convex object in Rd, and P a set of n points lying outside C. Further let cp, cq be two integers with 1 ≤ cq η cp ≤ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contains a subset of size cq + ⌊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 ℝd. For example, it follows from our theorem that when p > q ≥ (1 + β) · d/2 for β > 0, then any set of balls satisfying the HD(p, q) property can be hit by O (q2p1+1/β logp p) points. This is the first improvement over a nearly 60-year old exponential bound of roughly O(2d). 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 ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

Original languageEnglish (US)
Title of host publication26th European Symposium on Algorithms, ESA 2018
EditorsHannah Bast, Grzegorz Herman, Yossi Azar
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print)9783959770811
DOIs
StatePublished - Aug 1 2018
Event26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland
Duration: Aug 20 2018Aug 22 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume112
ISSN (Print)1868-8969

Other

Other26th European Symposium on Algorithms, ESA 2018
CountryFinland
CityHelsinki
Period8/20/188/22/18

Keywords

  • Balls
  • Convex polytopes
  • Epsilon-nets
  • Hadwiger-Debrunner numbers

ASJC Scopus subject areas

  • Software

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  • Cite this

    Mustafa, N. H., & Ray, S. (2018). On a problem of danzer. In H. Bast, G. Herman, & Y. Azar (Eds.), 26th European Symposium on Algorithms, ESA 2018 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 112). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2018.64