2026-10-19T04:45:20Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9527 2024-03-06T09:43:58Z ddc:004 open_access

On a Problem of Danzer Mustafa, Nabil H. Ray, Saurabh Convex polytopes Hadwiger-Debrunner numbers Epsilon-nets Balls 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 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+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 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Nabil H. Mustafa and Saurabh Ray 2018 Is Part Of LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2018.64 urn:nbn:de:0030-drops-95271 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.64 eng https://creativecommons.org/licenses/by/3.0/legalcode