Hyperspaces of Keller compacta and their orbit spaces

A compact convex subset K of a topological linear space is called a Keller compactum if it is affinely homeomorphic to an infinite-dimensional compact convex subset of the Hilbert space ℓ2. Let G be a compact topological group acting affinely on a Keller compactum K and let 2K denote the hyperspace...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2014-04, Vol.412 (2), p.613-619
Main Authors: Antonyan, Sergey A., Jonard-Pérez, Natalia, Juárez-Ordóñez, Saúl
Format: Article
Language:English
Subjects:
Affine group
Hyperspace
Infinite-dimensional convex set
Keller compactum
Orbit space
Q-manifold
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Summary:A compact convex subset K of a topological linear space is called a Keller compactum if it is affinely homeomorphic to an infinite-dimensional compact convex subset of the Hilbert space ℓ2. Let G be a compact topological group acting affinely on a Keller compactum K and let 2K denote the hyperspace of all non-empty compact subsets of K endowed with the Hausdorff metric topology and the induced action of G. Further, let cc(K) denote the subspace of 2K consisting of all compact convex subsets of K. In a particular case, the main result of the paper asserts that if K is centrally symmetric, then the orbit spaces 2K/G and cc(K)/G are homeomorphic to the Hilbert cube.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.10.076