  We present an alternative proof of the following fact: the hyperspace of
compact closed subsets of constant width in $\mathbb R^n$ is a contractible
Hilbert cube manifold. The proof also works for certain subspaces of compact
convex sets of constant width as well as for the pairs of compact convex sets
of constant relative width. Besides, it is proved that the projection map of
compact closed subsets of constant width is not 0-soft in the sense of
Shchepin, in particular, is not open.
