Hyperspaces of Keller compacta and their orbit spaces

A compact convex subset K of a topological linear space is called a Keller compactum if it is affinely homeomorphic to an infinite-dimensional compact convex subset of the Hilbert space ℓ2. Let G be a compact topological group acting affinely on a Keller compactum K and let 2K denote the hyperspace of all non-empty compact subsets of K endowed with the Hausdorff metric topology and the induced action of G. Further, let cc(K) denote the subspace of 2K consisting of all compact convex subsets of K. In a particular case, the main result of the paper asserts that if K is centrally symmetric, then the orbit spaces 2K/G and cc(K)/G are homeomorphic to the Hilbert cube.