On a Problem of Danzer

Nabil H. Mustafa, Saurabh Ray

Research output: Contribution to journalArticle

Abstract

Let C be a bounded convex object in d, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 1/2 cq 1/2 cp 1/2 n - d/2, such that every cp + d/2 points of P contain 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 hulls are 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 d. 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 (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) 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, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in d for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

Original languageEnglish (US)
Pages (from-to)1-10
Number of pages10
JournalCombinatorics Probability and Computing
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Polynomials
Convex Hull
Disjoint
Ball
Theorem
Partition
Exponential Bound
Imply
Polynomial
Subset
Hits
Convex Sets
Polynomial time
Union
Complement
Integer
Range of data
Object

Keywords

  • 2010 Mathematics subject classification:
  • 52C45

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

On a Problem of Danzer. / Mustafa, Nabil H.; Ray, Saurabh.

In: Combinatorics Probability and Computing, 01.01.2018, p. 1-10.

Research output: Contribution to journalArticle

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