Affine group acting on hyperspaces of compact convex subsets of \({\mathbb R}^{n}\)
The paper under review gives topological models for certain classical subspaces of the hyperspace \(cc(\mathbb{R}^n)\) of all compact convex nonempty subsets of \(\mathbb{R}^n\) equipped with the Hausdorff metric. The proofs are based on classical results of infinite-dimensional topology and on proper group actions and slices. In particular, the authors show that the group of affine isomorphisms of \(\mathbb{R}^n\) acts properly on the hyperspace \(cb(\mathbb{R}^n)\) of all compact convex bodies in \(\mathbb{R}^n\) (Theorem 3.3) and that its orbit space is homeomorphic to the Banach-Mazur compactum \(BM(n)\) of (all) real \(n\)-dimensional normed vector spaces (Corollary 3.8). Among many other results the following are worth mentioning: the hyperspace \(E(n)\) of all ellipsoids (in \(\mathbb{R}^n\)) is homeomorhic to \(\mathbb{R}^{n(n+3)/2}\) (Corollary 3.10) and \(cb(\mathbb{R}^n)\) is homeomorphic to \(Q \times \mathbb{R}^{n(n+3)/2}\) where \(Q\) is the Hilbert cube (Corollary 3.11); the hyperspace \(M(n)\) of all nonempty compact convex subsets of the unit ball that meet its boundary is homeomorphic to \(Q\) (Corollary 4.13); if \(K\) is a closed subgroup of the orthogonal group \(O(n)\) that acts nontransitively on the sphere around the origin, then the orbit spaces \(cb(\mathbb{R}^n) / K\) and \(cc(\mathbb{R}^n) / K\) are homeomorphic to, respectively, \((E(n) / K) \times Q\) (Theorem 6.1) and \(Q \setminus \{point\}\) (Theorem 7.1); the orbit space \(cc(\mathbb{R}^n) / O(n)\) is homeomorphic to the open cone over \(BM(n)\). The paper is concluded with a few interesting questions.
scientific article; zbMATH DE number 6238598