On critical values of numerical parameters characterizing intersections of embedded sets (Q5930798)

By an admissible sequence \(\{F_n\}\) in a Banach space \(X\) is meant a sequence \(F_1\supseteq F_2\supseteq\cdots\supseteq F_n\supseteq\cdots\) of embedded bounded closed subsets of \(X\). The main problem is to decide when the intersection \(\bigcap\{F_n: n\in\mathbb{N}\}\) is nonempty. With every subset \(F\subseteq X\) is associated the numerical parameter \(\kappa(F)\) defined by \(\kappa(F)= \sup_{x\in F}{r_x(F)\over R_x(F)}\), where \(R_x(F)= \sup_{y\in F}\|x-y\|\) and \(r_x(F)= \inf_{y\in F^c}\|x-y\|\). The author proves that for every \(\kappa\in [{1\over 3},{1\over 2}]\) there exists a Banach space \(X\) such that for every admissible sequence \(\{F_n\}\) in \(X\) the condition \(\varlimsup\kappa(F_n)> \kappa\) implies that the sequence \(\{F_n\}\) has a nonempty intersection, and for every \(\kappa_1< \kappa\) there is an admissible sequence \(\{F_n\}\) in \(X\) for which \(\kappa_1\leq\lim\kappa(F_n)< \kappa\) and the sequence \(\{F_n\}\) has empty intersection.