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

Abstract

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) (expX)/G is an ANR (resp., an AR). This is applied to show that (exp G)/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 subset of R-n, n greater than or equal to 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 subset of O(n), L-0(n)/H is a Hilbert cube manifold

(2) for each closed subgroup K subset of O(n) acting non-transitively on Sn-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 Sn-1) n {S(n-)1} have the same O(n)-homotopy type.