### Abstract

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} η c_{p} ≤ n - ⌊d/2⌋, such that every c_{p} + ⌊d/2⌋ points of P contains 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-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 + β) · d/2 for β > 0, then any set of balls satisfying the HD(p, q) property can be hit by O (q^{2}p^{1}+1/β logp 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.

Original language | English (US) |
---|---|

Title of host publication | 26th European Symposium on Algorithms, ESA 2018 |

Editors | Hannah Bast, Grzegorz Herman, Yossi Azar |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 112 |

ISBN (Print) | 9783959770811 |

DOIs | |

State | Published - Aug 1 2018 |

Event | 26th European Symposium on Algorithms, ESA 2018 - Helsinki, Finland Duration: Aug 20 2018 → Aug 22 2018 |

### Other

Other | 26th European Symposium on Algorithms, ESA 2018 |
---|---|

Country | Finland |

City | Helsinki |

Period | 8/20/18 → 8/22/18 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*26th European Symposium on Algorithms, ESA 2018*(Vol. 112). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2018.64

**On a problem of danzer.** / Mustafa, Nabil H.; Ray, Saurabh.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*26th European Symposium on Algorithms, ESA 2018.*vol. 112, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 26th European Symposium on Algorithms, ESA 2018, Helsinki, Finland, 8/20/18. https://doi.org/10.4230/LIPIcs.ESA.2018.64

}

TY - GEN

T1 - On a problem of danzer

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Balls

KW - Convex polytopes

KW - Epsilon-nets

KW - Hadwiger-Debrunner numbers

UR - http://www.scopus.com/inward/record.url?scp=85052516253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052516253&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2018.64

DO - 10.4230/LIPIcs.ESA.2018.64

M3 - Conference contribution

AN - SCOPUS:85052516253

SN - 9783959770811

VL - 112

BT - 26th European Symposium on Algorithms, ESA 2018

A2 - Bast, Hannah

A2 - Herman, Grzegorz

A2 - Azar, Yossi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -