The structure of the normed lattice generated by the closed, bounded, convex subsets of a normed space (Q2821981)

Let \(\mathcal C(X)\) denote the set of all non-empty closed bounded convex subsets of a normed linear space \(X\). In [Proc. Am. Math. Soc. 3, 165--169 (1952; Zbl 0046.33304)], \textit{H. Rådström} constructed a canonical isometric embedding of \(\mathcal{C}(X)\) equipped with the Hausdorff metric into a normed lattice \(R(X)\) with the order an extension of set inclusion in \(\mathcal{C}(X)\). In this paper, the authors initiate a study of the normed space structure of \(R(X)\) and its relation with the structure of \(X\). The main results are the following theorems:NEWLINENEWLINETheorem 1. If \(\dim X \geq 2\), then \(R(X)\) is incomplete.NEWLINENEWLINETheorem 2. Suppose \(Y\) is a subspace of \(X\), not necessarily closed, then \(R(Y )\) is isometrically isomorphic to a closed subspace of \(R(X)\). Moreover, if \(Y\) is dense in \(X\), then this subspace is the entirety of \(R(X)\).