Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan ; Natalia Jonard-Pérez

Fundamenta Mathematicae, Tome 220 (2013), p. 99-136 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Détail

Publié le : 2013-01-01

Zbl 1301.57018

EUDML-ID : urn:eudml:doc:282900