Allowable interval sequences and separating convex sets in the plane
In [Math. Scand. 45, 255--260 (1979; Zbl 0421.52004)], \textit{H. Tverberg} studied, for any given natural numbers \(r,k\), the existence of the smallest number of pairwise disjoint \(d\)-dimensional convex sets such that there is a closed half-space containing at least \(r\) of the sets, while the complementary closed half-space contains at least \(k\) of the remaining sets (this is usually called the \textit{(\(r,k\))-separation problem}). In the paper under review the author considers the case \(r=1\) in the plane and, denoting by \(f_k\) the above defined number, he improves previously known bounds by showing that \(f_k\leq \left\lceil(4+\sqrt{10})(k-1)-(3+\sqrt{10})/2\right\rceil\). In order to prove this result he applies the method of the so-called \textit{allowable interval sequences}, a kind of combinatorial object encoding properties of a family of pairwise disjoint planar convex sets, which was introduced and successfully used by Goodman and Pollack in 2005.
scientific article; zbMATH DE number 6010484