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

Heppes, Aladár

Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 388-402

Résumé

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.

Détail

DOI : 10.4153/CMB-2009-042-6

Mots-clés : 52A35, transversal, σ-disjoint, T(k)-family, Helly number, residue