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Discrete & Computational Geometry

, Volume 42, Issue 4, pp 594–614 | Cite as

Contraction and Expansion of Convex Sets

  • Michael Langberg
  • Leonard J. Schulman
Article

Abstract

Let \({\mathcal{S}}\) be a set system of convex sets in ℝ d . Helly’s theorem states that if all sets in \({\mathcal{S}}\) have empty intersection, then there is a subset \({\mathcal{S}}'\subset{\mathcal{S}}\) of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in \({\mathcal{S}}\) are not convex or if \({\mathcal{S}}\) does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets.

These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C ε and the expansion C ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:

(a) If \({\mathcal{S}}\) is any family of convex sets in ℝ d , then there is a finite subfamily \({\mathcal{S}}'\subseteq{\mathcal{S}}\) whose cardinality depends only on ε and d, such that \(\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon}\subseteq\bigcap_{C\in {\mathcal{S}}}C\) .

The second result allows the sets in \({\mathcal{S}}\) a limited type of nonconvexity:

(b) If \({\mathcal{S}}\) is a family of sets in ℝ d , each of which is the union of k fat convex sets, then there is a finite subfamily \({\mathcal{S}}'\subseteq{\mathcal{S}}\) whose cardinality depends only on ε, d, and k, such that \(\bigcap_{C\in{\mathcal{S}}'}C^{-\varepsilon }\subseteq \bigcap_{C\in{\mathcal{S}}}C\) .

Keywords

Helly-type theorems Nonconvex 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Computer Science DivisionThe Open University of IsraelRaananaIsrael
  2. 2.Department of Computer ScienceCalifornia Institute of TechnologyPasadenaUSA

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