Canadian mathematical bulletin | |
Transversals with Residue in Moderately Overlapping $T(k)$-Families of Translates | |
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
【 预 览 】
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