Discrete & Computational Geometry

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

Contraction and Expansion of Convex Sets

Article

Abstract

Let \({\mathcal{S}}\) be a set system of convex sets in ℝd. 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 Cε and the expansion Cε 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 ℝd, 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 ℝd, each of which is the union of kfat 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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