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On a Problem of Danzer

Part of: Discrete geometry

Published online by Cambridge University Press:  15 October 2018

NABIL H. MUSTAFA  and
SAURABH RAY
NABIL H. MUSTAFA
Affiliation:
Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, Equipe A3SI, ESIEE Paris, France (e-mail: [email protected])
SAURABH RAY
Affiliation:
Computer Science, New York University, Abu Dhabi, United Arab Emirates (e-mail: [email protected])
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Abstract

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ 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 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 (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 ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

MSC classification

Primary: 52C45: Combinatorial complexity of geometric structures
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 473 - 482
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†The conference version of this paper appeared in the 26th Annual European Symposium on Algorithms, 2018.

‡

The work of Nabil H. Mustafa in this paper was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).

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