Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning View Full Text


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Article Info

DATE

2019-03-15

AUTHORS

Kunal Dutta, Arijit Ghosh, Bruno Jartoux, Nabil H. Mustafa

ABSTRACT

Given a set system (X,R) such that every pair of sets in R have large symmetric difference, the Shallow Packing Lemma gives an upper bound on |R| as a function of the shallow-cell complexity of R. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system (X,R) with shallow-cell complexity φ(·,·) and a parameter ϵ>0, there exists a collection, called an ϵ-Mnet, consisting of O(1ϵφ(O(1ϵ),O(1))) subsets of X, each of size Ω(ϵ|X|), such that any R∈R with |R|≥ϵ|X| contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal ϵ-net bound follows. More... »

PAGES

1-22

References to SciGraph publications

  • 1999. Geometric Discrepancy, An Illustrated Guide in NONE
  • 1989-10. Applications of random sampling in computational geometry, II in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2016-12. Two Proofs for Shallow Packings in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2007. Stochastic Geometry, Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004 in NONE
  • 2015-07. Multilevel Polynomial Partitions and Simplified Range Searching in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2016-04. A Simple Proof of the Shallow Packing Lemma in DISCRETE & COMPUTATIONAL GEOMETRY
  • 2013. Topological Hypergraphs in THIRTY ESSAYS ON GEOMETRIC GRAPH THEORY
  • 2002. Lectures on Discrete Geometry in NONE
  • 2017. Near-Optimal Lower Bounds for ε-Nets for Half-Spaces and Low Complexity Set Systems in A JOURNEY THROUGH DISCRETE MATHEMATICS
  • 2017-04. ε-Mnets: Hitting Geometric Set Systems with Subsets in DISCRETE & COMPUTATIONAL GEOMETRY
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00454-019-00075-0

    DOI

    http://dx.doi.org/10.1007/s00454-019-00075-0

    DIMENSIONS

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