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

Canadian mathematical bulletin

关键词: transversal;    $sigma$-disjoint;    $T(k)$-family;    Helly number;    residue;   

DOI  :  10.4153/CMB-2009-042-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $K$ denote an oval, a centrally symmetric compact convex domainwith non-empty interior. A family of translates of $K$ is said to haveproperty $T(k)$ if for every subset of at most $k$ translates thereexists 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 $k(sigma)$ such that if afamily of $sigma$-disjoint unit diameter discs has property $T(k)| kgeq k(sigma)$, then there exists a straight line meeting all membersof the family. In the first part of the paper we give the extension ofthis theorem to translates of an oval $K$. The asymptotic behavior of$k(sigma)$ for $sigma ightarrow 0$ is considered as well.Katchalski and Lewis proved the existence of a constant $r$ such thatfor every pairwise disjoint family of translates of an oval $K$ withproperty $T(3)$ 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 therelation of $sigma$ and the residue $r$ is investigated. Theasymptotic behavior of $r(sigma)$ for $sigma ightarrow 0$ is alsodiscussed.

【 授权许可】

Unknown