Piercing convex sets and the Hadwiger-Debrunner \((p,q)\)-problem
Let \(p\), \(q\), \(d\) be natural numbers with \(p\geq q\geq d+1\). Then, a \(c=c(p,q,d)<\infty\) exists with the property: for each family \(\mathcal F\) of compact convex sets in \(\mathbb{R}^ d\) with the \((p,q)\)-property (i.e., among any \(p\) members of \(\mathcal F\) some \(q\) have a nonempty intersection) there is a set of at most \(c\) points in \(\mathbb{R}^ n\) that intersects each member of \(\mathcal F\). This result settles an old problem of \textit{H. Hadwiger} and \textit{H. Debrunner} [Arch. Math. 8, 309-313 (1957; Zbl 0080.154)].
scientific article; zbMATH DE number 149037
scientific article