Discrete & Computational Geometry

, Volume 42, Issue 4, pp 594–614

# 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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