Recognizing the topology of the space of closed convex subsets of a Banach space

Details Recognizing the topology of the space of closed convex subsets of a Banach space Recognizing the topology of the space of closed convex subsets of a Banach space

2011-12-29

arxiv.org/abs/1112.6374v1

Mathematics

Geometric Topology

10 pages

Scientific paper

Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by the half-line, the separable Hilbert space, or a Hilbert space of density not less than continuum.

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