# Contraction and expansion of convex sets

Michael Langberg, Leonard J. Schulman

نتاج البحث: نتاج بحثي من مؤتمرمحاضرةمراجعة النظراء

## ملخص

Let S be a set system of convex sets in ℝd. Helly's theorem states that if all sets in S have empty intersection, then there is a subset S′ ⊂ S of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S are not convex or if 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 and the expansion Cε are close (in Hausdorff) to C. We obtain two results. The first extends Helly's theorem to the case of set systems with non-empty intersection: (a) If S is any family of convex sets in ℝd then there is a finite subfamily S′ ⊆ S whose cardinality depends only on ε and d, such that ∩C∈S′C ⊆∩C∈SC. The second result allows the sets in S a limited type of non-convexity: (b) If S is a family of sets in ℝd, each of which is the union of k fat convex sets, then there is a finite subfamily S′ ⊆; S whose cardinality depends only on ε, d and k, such that ∩C∈S′ ⊆∩ C∈SC.

اللغة الأصلية الإنجليزيّة 25-28 4 نُشِر - 2007 19th Annual Canadian Conference on Computational Geometry, CCCG 2007 - Ottawa, ON, كنداالمدة: ٢٠ أغسطس ٢٠٠٧ → ٢٢ أغسطس ٢٠٠٧

### !!Conference

!!Conference 19th Annual Canadian Conference on Computational Geometry, CCCG 2007 كندا Ottawa, ON ٢٠/٠٨/٠٧ → ٢٢/٠٨/٠٧

## بصمة

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