### Abstract

Let C be a bounded convex object in ^{d}, and let P be a set of n points lying outside C. Further, let c_{p}, c_{q} be two integers with 1 1/2 c_{q} 1/2 c_{p} 1/2 n - d/2, such that every c_{p} + d/2 points of P contain a subset of size c_{q} + 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)^{2}d^{2}p^{1+1/β} logp) 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, 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 language | English (US) |
---|---|

Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Combinatorics Probability and Computing |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### Keywords

- 2010 Mathematics subject classification:
- 52C45

### ASJC Scopus subject areas

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

### Cite this

*Combinatorics Probability and Computing*, 1-10. https://doi.org/10.1017/S0963548318000445

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

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, pp. 1-10. https://doi.org/10.1017/S0963548318000445

}

TY - JOUR

T1 - On a Problem of Danzer

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - 2010 Mathematics subject classification:

KW - 52C45

UR - http://www.scopus.com/inward/record.url?scp=85054989044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054989044&partnerID=8YFLogxK

U2 - 10.1017/S0963548318000445

DO - 10.1017/S0963548318000445

M3 - Article

SP - 1

EP - 10

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

ER -