The Banach–Mazur compactum is homeomorphic to the orbit space

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 ( exp S 1 ) / O ( 2 ) , where exp S 1 is the hyperspace of all nonempty closed subsets of the unit circle S 1 endowed with the induced action of the orthogonal group O ( 2 ) .