Contraction and expansion of convex sets
The authors introduce two operations on convex sets, called contraction and expansion, that generalize scaling of centrally symmetric sets. The operations are defined for \(C\) convex and \(\varepsilon \geq -1\) using intersections of slabs. With the help of these new operations, the authors prove two Helly-type theorems: one for families of sets that have empty intersection, and one for families of sets in which the members are not convex sets of a certain limited type. More precisely, with \(C^\varepsilon\) being the expansion and \(C^{-\varepsilon}\) being the contraction of the convex set \(C\) with parameter \(\varepsilon\), they establish the following two statements. {\parindent6mm \begin{itemize}\item[(a)] If \({\mathcal S}\) is a family of convex sets in \({\mathbb R}^d\), then there is a finite subfamily \({\mathcal S}'\subseteq {\mathcal S}\) whose cardinality depends only on \(\varepsilon\) and \(d\), such that \(\bigcap_{C\in{\mathcal S}'}C^{-\varepsilon}\subseteq\bigcap_{C\in {\mathcal S}}C\). \end{itemize}} For a real number \(k\geq 1\), we call a convex set \(C\) \(k\)-fat, if the ratio of the radius of the circumsphere of \(C\) and the radius of the insphere of \(C\) is equal to \(k\). {\parindent6mm \begin{itemize}\item[(b)] If \({\mathcal S}\) is a family of sets in \({\mathbb R}^d\), each of which is the union of \(k\)-fat convex sets, then there is a finite subfamily \({\mathcal S}'\subseteq {\mathcal S}\) whose cardinality depends only on \(\varepsilon\), \(d\), and \(k\), such that \(\bigcap_{C\in{\mathcal S}'}C^{-\varepsilon}\subseteq\bigcap_{C\in {\mathcal S}}C\). \end{itemize}}
scientific article; zbMATH DE number 5646270