Topological structure of the hyperspace of full-dimensional closed convex subsets of R^n with the Fell topology

Determine the topological structure (e.g., homeomorphism type or a complete classification) of the hyperspace (K_n, TF), where K_n denotes the subspace of K^n consisting of all closed convex subsets of R^n whose affine hull has dimension n (equivalently, closed convex subsets with nonempty interior), and TF denotes the Fell topology.

Background

The paper studies hyperspaces of closed convex subsets of a Hilbert space, focusing on finite-dimensional Euclidean spaces R^n. It uses several hyperspace topologies, notably the Fell topology (TF) and the Attouch–Wets topology, and analyzes subspaces defined by dimension.

Let K^n be the hyperspace of all nonempty closed convex subsets of R^n, and K_n the subspace of those with full affine dimension n (i.e., nonempty interior). While it is known that (K^n, TF) is homeomorphic to R^n × Q (where Q is the Hilbert cube), the precise topological type of (K_n, TF) is not established in the literature.

In this paper, Proposition 2.11 shows that (K_n, TF) is an open subspace of (K^n, TF) and thus a Q-manifold. However, the authors explicitly note that the overall topological structure of (K_n, TF) remains unknown, indicating a gap in the current classification beyond the Q-manifold property.