Recognizing the topology of the space of closed convex subsets of a Banach space (Q2839267)

In this interesting paper the authors topologically characterize the space of all nonempty closed convex subsets of a Banach space \(B\) endowed with the Hausdorff metric. In fact they show that if \(\mathcal{H}\) is a component of the hyperspace under consideration, then \(\mathcal{H}\) is (topologically) \(\{ 0\}\) iff \(\mathcal{H}\) contains \(B\), \(\mathbb{R}\) iff \(\mathcal{H}\) contains a half-space of \(B\), \(\mathbb{R}\times [0,\infty)\) iff \(\mathcal{H}\) contains a linear subspace of codimension 1 of \(B\), \(Q\times [0,\infty)\) iff \(\mathcal{H}\) contains a linear subspace of codimension \({>}1\) of \(B\) (here \(Q\) denotes the Hilbert cube), a separable Hilbert space iff \(\mathcal{H}\) contains a polyhedral convex subset of \(B\) but contains no linear subspace and no half-space of \(B\), a Hilbert space of weight at least continuum iff \(\mathcal{H}\) contains no polyhedral convex subspace of \(B\). The proof is based among other things on some interesting geometrical observations and the result of Banakh and Cauty that each non-locally compact closed convex subset of a Banach space is homeomorphic to a Hilbert space. Banach spaces with the Kunen-Shelah property are also considered.