Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning

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

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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.

Original languageEnglish
JournalDiscrete and Computational Geometry
DOIs
StateAccepted/In press - 1 Jan 2019
Externally publishedYes

Keywords

  • Epsilon-nets
  • Haussler’s Packing Lemma
  • Mnets
  • Shallow Packing Lemma
  • Shallow-cell complexity

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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