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

TL;DR

This paper characterizes the topological structure of the space of non-empty closed convex subsets in a Banach space, revealing it consists of components homeomorphic to well-known topological spaces.

Contribution

It provides a complete topological classification of connected components of the space of convex subsets in Banach spaces, identifying their homeomorphism types.

Findings

Connected components are homeomorphic to specific classical spaces.

The space includes components homeomorphic to the Hilbert cube, Hilbert spaces, and simpler spaces.

The classification depends on the properties of the Banach space.

Abstract

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.