Wide and tight spherical hulls of bounded sets in Banach spaces
For a Banach space \(X\), let \(\mathcal H\) denote the class of bounded closed convex non-singleton subsets of \(X\). A set \(A \in \mathcal H\) is said to be complete provided that the diameter of \(A \cup \{ x\}\) is strictly larger than the diameter of \(A\) whenever \(x \notin A\). For any \(A \in \mathcal H\), a completion of \(A\) is any complete set \(B\) (clearly still in \(\mathcal H\)) containing \(A\) and having the same diameter as \(A\). For fixed \(A \in \mathcal H\), this paper deals with the sets \(\eta (A)\), union of the completions of \(A\), and \(\theta (A)\), intersection of the completions of \(A\), respectively called wide and tight spherical hulls of \(A\). If, for \(A \in \mathcal H\) and any \(x \in X\), \(\gamma (A,x)\) denotes the supremum of \(\| x-a \|\) as \(a\) ranges in \(A\), the main results are the following: {\parindent =0.4 cm \begin{itemize}\item[--] \(x \in \partial \eta (A)\) if and only if \(\gamma (A,x)\) equals the diameter of \(A\) (Proposition 2.1); \item[--] for \(x \in \theta (A)\), \(x \in \partial \theta (A)\) if and only if \(\gamma (\eta (A),x)\) equals the diameter of \(A\) (Proposition 2.2); \item[--] a point in \(A\) is a diametral point of \(A\) if and only if it is a diametral point of \(\theta (A)\) (Proposition 2.6); \item[--] the distance of \(A\) to the boundary of each of its spherical hull is \(0\) (though the intersection of \(A\) with those boundaries may be empty) (Propositions 2.9 and 2.11); \item[--] \(A \in \mathcal H\) is a ball if and only if it is the only completion of some element of \(\mathcal H\) and the diameter of \(A\) is the infimum of \(\gamma (A,x)\) as \(x\) ranges in \(X\) (Corollary 3.3). \end{itemize}} The results are based on basic properties of the spherical hulls shown in some papers quoted in the references. Significant suitable examples and counterexamples are provided.
scientific article; zbMATH DE number 6575542