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Let G be a compact Lie group, X a metric G-space, and exp X the hyperspace of all nonempty compact subsets of X endowed with the Hausdorff metric topology and with the induced action of G. We prove that the following three assertions are equivalent: (a) X is locally continuum-connected (resp., connected and locally continuum-connected); (b) expX is a G-ANR (resp., a G-AR); (c) (exp X)/G is an ANR (resp., an AR). This is applied to show that (expG)/G is an ANR (resp., an AR) for each compact (resp., connected) Lie group G. If G is a finite group, then (expX)/G is a Hilbert cube whenever X is a nondegenerate Peano continuum. Let L(n) be the hyperspace of all centrally symmetric, compact, convex bodies A C R n , n > 2, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing A, and let L 0 (n) be the complement of the unique O(n)-fixed point in L(n). We prove that: (1) for each closed subgroup H C O(n), L 0 (n)/H is a Hilbert cube manifold; (2) for each closed subgroup K C O(n) acting non-transitively on S n-1 , the K-orbit space L(n)/K and the K-fixed point set L(n)[K] are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta L(n)/O(n) and prove that L 0 ( n ) and (exp S n-1 ) \ {S n-1 } have the same O(n)-homotopy type.

West’s problem on equivariant hyperspaces and Banach-Mazur compacta