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 ℓ 2 . Let G be a compact topological group acting affinely on a Keller compactum K and let 2 K 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 c c ( K ) denote the subspace of 2 K 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 2 K / G and c c ( K ) / G are homeomorphic to the Hilbert cube.