Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan, and Natalia Jonard-Pérez. "Affine group acting on hyperspaces of compact convex subsets of ℝⁿ." Fundamenta Mathematicae 223.2 (2013): 99-136. <http://eudml.org/doc/282900>.

@article{SergeyA2013,
abstract = {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).},
author = {Sergey A. Antonyan, Natalia Jonard-Pérez},
journal = {Fundamenta Mathematicae},
keywords = {convex set; hyperspace; affine group; proper action; slice; orbit space; Banach-Mazur compactum; Q-manifold},
language = {eng},
number = {2},
pages = {99-136},
title = {Affine group acting on hyperspaces of compact convex subsets of ℝⁿ},
url = {http://eudml.org/doc/282900},
volume = {223},
year = {2013},
}

TY - JOUR
AU - Sergey A. Antonyan
AU - Natalia Jonard-Pérez
TI - Affine group acting on hyperspaces of compact convex subsets of ℝⁿ
JO - Fundamenta Mathematicae
PY - 2013
VL - 223
IS - 2
SP - 99
EP - 136
AB - 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).
LA - eng
KW - convex set; hyperspace; affine group; proper action; slice; orbit space; Banach-Mazur compactum; Q-manifold
UR - http://eudml.org/doc/282900
ER -