Topology of the hyperspace of convex bodies of constant width

The hyperspace of all convex bodies of constant width in Euclidean space $\mathbb R^n$, $n\ge2$, is proved to be homeomorphic to a contractible $Q$-manifold ($Q$ denotes the Hilbert cube). The proof makes use of an explicitly constructed retraction of the entire hyperspace of convex bodies on the hyperspace of convex bodies of constant width.