Abstract view
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Transversals with Residue in Moderately Overlapping $T(k)$-Families of Translates
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Published:2009-09-01
Printed: Sep 2009
Abstract
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 $T(k)$ 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 $k(\sigma)$ such that if a
family of $\sigma$-disjoint unit diameter discs has property $T(k)| k
\geq k(\sigma)$, 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
$k(\sigma)$ for $\sigma \rightarrow 0$ 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 $T(3)$ 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 the
relation of $\sigma$ and the residue $r$ is investigated. The
asymptotic behavior of $r(\sigma)$ for $\sigma \rightarrow 0$ is also
discussed.
