Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

• Autorzy: Sergey A. Antonyan , Natalia Jonard-Pérez
• Języki publikacji: EN

Abstrakty

EN

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 $𝕊^{n-1}$, 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).

Kategorie tematyczne

• 46B99: None of the above, but in this section
• 57S10: Compact groups of homeomorphisms
• 55P91: Equivariant homotopy theory
• 54B20: Hyperspaces
• 57N20: Topology of infinite-dimensional manifolds
• 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 223
• Numer: 2
• Strony: 99-136

Daty

wydano

2013

Twórcy

autor

Sergey A. Antonyan

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

autor

Natalia Jonard-Pérez

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

Identyfikatory

DOI

10.4064/fm223-2-1