Hyperspaces of convex bodies of constant width

For every n≥1, let cc(Rn) denote the hyperspace of all non-empty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric topology. For every non-empty convex subset D of [0,∞) we denote by cwD(Rn) the subspace of cc(Rn) consisting of all compact convex sets of constant width d∈D and by crwD(Rn) the subspace of the product cc(Rn)×cc(Rn) consisting of all pairs of compact convex sets of constant relative width d∈D. In this paper we prove that cwD(Rn) and crwD(Rn) are homeomorphic to D×Rn×Q, whenever D≠{0} and n≥2, where Q denotes the Hilbert cube. In particular, the hyperspace cw(Rn) of all compact convex bodies of constant width as well as the hyperspace crw(Rn) of all pairs of compact convex sets of constant relative positive width are homeomorphic to Rn+1×Q.