Affine group acting on hyperspaces of compact convex subsets of ${\mathbb R}^{n}$
Volume 223 / 2013
Abstract
For every $n\ge 2$, let $cc(\mathbb R^{n})$ denote the hyperspace of all nonempty compact convex subsets of the Euclidean space $\mathbb R^n$ endowed with the Hausdorff metric topology. Let $cb(\mathbb R^{n})$ be the subset of $cc(\mathbb R^{n})$ consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group $\mathop {\rm Aff}(n)$ on $cb(\mathbb R^{n})$. We prove that the space $E(n)$ of all $n$-dimensional ellipsoids is an $\mathop {\rm Aff}(n)$-equivariant retract of $cb(\mathbb R^{n})$. This is applied to show that $cb(\mathbb R^{n})$ is homeomorphic to the product $Q\times \mathbb R^{n(n+3)/2}$, where $Q$ stands for the Hilbert cube. Furthermore, we investigate the action of the orthogonal group $O(n)$ on $cc(\mathbb R^{n})$. In particular, we show that if $K\subset O(n)$ is a closed subgroup that acts nontransitively on the unit sphere $\mathbb S^{n-1}$, then the orbit space $cc(\mathbb R^{n})/K$ is homeomorphic to the Hilbert cube with a point removed, while $cb(\mathbb R^{n})/K$ is a contractible $Q$-manifold homeomorphic to the product $(E(n)/K)\times Q$. The orbit space $cb(\mathbb R^{n})/{\rm Aff}(n)$ is homeomorphic to the Banach–Mazur compactum ${\rm BM}(n)$, while $cc(\mathbb R^{n})/O(n)$ is homeomorphic to the open cone over ${\rm BM}(n)$.
