  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.
