Some remarks on the intersection of embedded sets
For a nonempty bounded subset \(A\) of an infinite-dimensional Banach space \(X\) and \(x\in X\), set \(\delta(A)=\) the diameter of the set \(A\), \(r(A,x)=\sup\{\| x-y\| : y\in A\}\), \(r'(A,x)=\inf\{\| x-y\| : y\in X\setminus A\}\), \(r(A)=\inf\{r(A,x) : x\in X\}\), \(r'(A)=\sup\{r'(A,x) : x\in X\}=\sup\{r'(A,x) : x\in A\}\), and \(\chi(A)=\sup\{r'(A,x)/r(A,x) : x\in A\}\). A sequence \(\{A_n\}\) of nonempty bounded closed subsets of \(X\) is called admissible if \(A_1\supset A_2\supset\dots\). The main problem that the authors address in the present paper is: what conditions on \(\chi(\{A_n\}):=\limsup_n \chi(A_n)\) ensure that \(\bigcap_nA_n\neq\emptyset\)? The critical value cv\((X)\) of the Banach space \(X\) is the number \(\alpha\geq 0\) such that \(\bigcap_nA_n\neq \emptyset\) for every admissible sequence \(\{A_n\}\) in \(X\) with \(\chi(\{A_n\})>\alpha,\) and for every \(\varepsilon >0\) there exists an admissible sequence \(\{A_n\}\) in \(X\) such that \(\chi(\{A_n\})>\alpha-\varepsilon\) and \(\bigcap_nA_n=\emptyset.\) It is known that cv\((X)\in [1/3,1/2]\) and that \(\chi(\ell^p)=1/(1+2^{1/p}), \) so that for every \(\alpha\in (1/3,1/2) \) there exists a reflexive Banach space with cv\((X)=\alpha.\) The authors show that there exists a reflexive Banach space \(X\) with cv\((X)=1/2,\) and that in a reflexive Banach space \(X\), \(\bigcap_nA_n\neq \emptyset\) for every admissible sequence \(\{A_n\}\) with \(\chi(\{A_n\})=\text{cv}(X).\) The existence of a reflexive Banach space \(X\) with cv\((X)=1/3\) is an open question. Putting \(\Delta(\{A_n\})=\limsup_nr'(A_n)/\delta(A_n)\), one defines in a similar way another characteristic value CV\((X)\), which is related to the previous one by the formula CV\((X)= \text{cv}(X)/(1+\text{cv}(X))\). The authors establish some interesting connections with the Jung constant \(J(X)=\sup\{2r(A)/\delta(A)\},\) where the supremum is taken over all nonempty bounded closed subsets \(A\) of \(X.\) It is shown that CV\((X)\leq J(X)/4\) and hence cv\((X)\leq J(X)/(4-J(X)).\)
scientific article; zbMATH DE number 5155543