Discrete & Computational Geometry

, Volume 42, Issue 3, pp 379–398 | Cite as

Helly-Type Theorems for Approximate Covering

  • Julien Demouth
  • Olivier Devillers
  • Marc Glisse
  • Xavier Goaoc
Article
  • 75 Downloads

Abstract

Let ℱ∪{U} be a collection of convex sets in ℝd such that ℱ covers U. We prove that if the elements of ℱ and U have comparable size, then a small subset of ℱ suffices to cover most of the volume of U; we analyze how small this subset can be depending on the geometry of the elements of ℱ and show that smooth convex sets and axis parallel squares behave differently. We obtain similar results for surface-to-surface visibility amongst balls in three dimensions for a notion of volume related to form factor. For each of these situations, we give an algorithm that takes ℱ and U as input and computes in time O(|ℱ|*|ℋε|) either a point in U not covered by ℱ or a subset ℋε covering U up to a measure ε, with |ℋε| meeting our combinatorial bounds.

Keywords

Approximate covering Helly-type theorems 3D visibility LP-type problems 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Julien Demouth
    • 1
  • Olivier Devillers
    • 2
  • Marc Glisse
    • 3
  • Xavier Goaoc
    • 1
  1. 1.LORIA—Université Nancy 2Villers-lès-NancyFrance
  2. 2.INRIA Sophia-AntipolisSophia-AntipolisFrance
  3. 3.Gipsa-Lab, CNRS UMR 5216Domaine UniversitaireSt Martin d’HèresFrance

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