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 <= c_p <= n  floor[d/2], such that every c_p + floor[d/2] points of P contains a subset of size c_q + floor[d/2] whose convexhull 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 convexhull 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 HadwigerDebrunner (p, q) numbers for balls in R^d. For example, it follows from our theorem that when p > q >= (1+beta) * d/2 for beta > 0, then any set of balls satisfying the HD(p,q) property can be hit by O(q^2 p^{1+1/(beta)} log p) points. This is the first improvement over a nearly 60year 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.
BibTeX  Entry
@InProceedings{mustafa_et_al:LIPIcs:2018:9527,
author = {Nabil H. Mustafa and Saurabh Ray},
title = {{On a Problem of Danzer}},
booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)},
pages = {64:164:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770811},
ISSN = {18688969},
year = {2018},
volume = {112},
editor = {Yossi Azar and Hannah Bast and Grzegorz Herman},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9527},
URN = {urn:nbn:de:0030drops95271},
doi = {10.4230/LIPIcs.ESA.2018.64},
annote = {Keywords: Convex polytopes, HadwigerDebrunner numbers, Epsilonnets, Balls}
}
Keywords: 

Convex polytopes, HadwigerDebrunner numbers, Epsilonnets, Balls 
Collection: 

26th Annual European Symposium on Algorithms (ESA 2018) 
Issue Date: 

2018 
Date of publication: 

14.08.2018 