Affine group acting on hyperspaces of compact convex subsets of ${\mathbb R}^{n}$

Sergey A. Antonyan ¹   ; Natalia Jonard-Pérez ¹

¹ Departamento de Matemáticas Facultad de Ciencias Universidad Nacional Autónoma de México 04510 México Distrito Federal, México

Fundamenta Mathematicae, Tome 223 (2013) no. 2, pp. 99-136

Résumé

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 ${\rm BM}(n)$.

Détail

DOI : 10.4064/fm223-2-1

Keywords: every mathbb denote hyperspace nonempty compact convex subsets euclidean space mathbb endowed hausdorff metric topology mathbb subset mathbb consisting compact convex bodies paper discover several fundamental properties natural action affine group mathop aff mathbb prove space n dimensional ellipsoids mathop aff equivariant retract mathbb applied mathbb homeomorphic product times mathbb where stands hilbert cube furthermore investigate action orthogonal group mathbb particular subset closed subgroup acts nontransitively unit sphere mathbb n orbit space mathbb homeomorphic hilbert cube point removed while mathbb contractible q manifold homeomorphic product times orbit space mathbb aff homeomorphic banach mazur compactum while mathbb homeomorphic cone