On convex bodies of constant width

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.