The Banach–Mazur compactum BM(2) is homeomorphic to the orbit space (expS1)/O(2)

We prove a general theorem about the orbit spaces of compact Lie group actions which are Hilbert cube manifolds. This result is further applied to prove that the Banach–Mazur compactum BM(2) is homeomorphic to the orbit space (expS1)/O(2), where expS1 is the hyperspace of all nonempty closed subsets of the unit circle S1 endowed with the induced action of the orthogonal group O(2).