The dimension of the kernel in finite and infinite intersections of starshaped sets (Q1881598)

Let \(\mathcal K\) be a uniformly bounded family of compact sets in \(\mathbb{R}^{2}\) and \(f\) be a function of the variable \(k\) which takes values in the set \(\{1,2\}\) and is defined by \(f(1)=4\), \(f(2)=3\). Making use of results of \textit{N. A. Bobylev} [J. Math. Sci., New York 105, No. 2, 1819--1825 (2001; Zbl 1013.52006)] and of a previous paper by herself [Arch. Math. 81, No. 4, 485--490 (2003; Zbl 1044.52004)], the author proves a Helly type theorem for infinite families of compact sets, viz. that if the intersection of every finite subset of \(\mathcal K\) having \(f(k)\) (not necessarily distinct) members, is a starshaped set whose kernel contains a \(k\)-dimensional neighborhood of radius \(\varepsilon \) (where \(\varepsilon \) is a fixed positive number) then \(\bigcap \{K:K\) in \( \mathcal{K}\}\) is a starshaped set whose kernel is at least \(k\)-dimensional. The author further proves that in each case the number \(f(k)\) is best possible choice.