A Helly-type theorem for countable intersections of starshaped sets
Let \(S\) be a set in \(\mathbb{R}^{d}\). For every pair of points \(x\) and \(y\) in \(S\), \(x\) is visible from \(y\) via \(S\) if and only if the line segment \([x,y]\) lies in \(S\). A set \(S\) is called starshaped if and only if for some point \(p\) in \(S\), \(p\) sees each point of \(S\) via \(S\) and the set of all such points \(p\) is the (convex) kernel of \(S\). In the first theorem taking \(k\) and \(d\) as fixed integers \(0\leq k\leq d\) and \(\mathcal{K} =\{K_{\alpha }:\alpha \) in some index set\(\}\) as a collection of sets in \( \mathbb{R}^{d}\) the author proves that if every countable subfamily of \(\mathcal{K}\) has a starshaped intersection whose kernel is at least \(k\)-dimensional then \(\bigcap \{K_{\alpha }:K_{\alpha }\) in \(\mathcal{K}\}\) also is a starshaped set whose kernel is at least \(k\)-dimensional. In the second theorem, again taking \(k\) and \(d\) as fixed integers \(0\leq k\leq d\) and \(\mathcal{K} =\{K_{\alpha }:\alpha \) in some index set\(\}\) as a collection of sets in \( \mathbb{R}^{d}\) it is proved that if every countable subfamily of \(\mathcal{K}\) has a starshaped union then \(\bigcup \{K_{\alpha }:K_{\alpha }\) in \(\mathcal{K}\}\) also is a starshaped set whose kernel is at least \(k\)-dimensional.
scientific article; zbMATH DE number 2162320