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Transversals with Residue in Moderately Overlapping T(k)-Families of Translates

Published online by Cambridge University Press:  20 November 2018

Aladár Heppes
Aladár Heppes*
Affiliation:
Rényi Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary e-mail: aheppes@renyi.hu
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Abstract

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Let $K$ denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of $K$ is said to have property $T\left( k \right)$ if for every subset of at most $k$ translates there exists a common line transversal intersecting all of them. The integer $k$ is the stabbing level of the family. Two translates ${{K}_{i}}\,=\,K\,+\,{{c}_{i}}$ and ${{K}_{j}}\,=\,K\,+\,{{c}_{j}}$ are said to be $\sigma$-disjoint if $\sigma K\,+\,{{c}_{i}}$ and $\sigma K\,+\,{{c}_{j}}$ are disjoint. A recent Helly-type result claims that for every $\sigma \,>\,0$ there exists an integer $k\left( \sigma \right)$ such that if a family of $\sigma$-disjoint unit diameter discs has property $T\left( k \right)|k\ge k\left( \sigma \right)$, then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval $k$. The asymptotic behavior of $k\left( \sigma \right)$ for $\sigma \,\to \,0$ is considered as well.

Katchalski and Lewis proved the existence of a constant $r$ such that for every pairwise disjoint family of translates of an oval $K$ with property $T\left( 3 \right)$ a straight line can be found meeting all but at most $r$ members of the family. In the second part of the paper $\sigma$-disjoint families of translates of $K$ are considered and the relation of $\sigma$ and the residue $r$ is investigated. The asymptotic behavior of $r\left( \sigma \right)$ for $\sigma \,\to \,0$ is also discussed.

Keywords

52A35transversalσ-disjointT(k)-familyHelly numberresidue
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 3 , 01 September 2009 , pp. 388 - 402
Copyright
Copyright © Canadian Mathematical Society 2009

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