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) $\exp X$ is a $G$-ANR (resp., a $G$-AR); (c) $(\exp X)/G$ 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 $(\exp X)/G$ is a Hilbert cube whenever $X$ is a nondegenerate Peano continuum. Let $L(n)$ be the hyperspace of all centrally symmetric, compact, convex bodies $A\subset \mathbb {R}^n$, $n\ge 2$, 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 $H\subset O(n)$, $L_0(n)/H$ is a Hilbert cube manifold; (2) for each closed subgroup $K\subset O(n)$ 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 $(\exp S^{n-1})\setminus \{S^{n-1}\}$ have the same $O(n)$-homotopy type.