  Let n be a natural number equal or greater than 2. In this paper we study the
topological structure of certain hyperspaces of convex subsets of constant
width, equipped with the Hausdorff metric topology. We focus our attention on
the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in
D, where D is a convex subset of [0,\infty). Our main result states that
cw_D(R^n) is homeomorphic to D\times R^n\times Q, where Q denotes the Hilbert
cube. We also prove that the hyperspace crw_D(R^n), consisting of all pairs of
compact convex sets of constant relative width d\in D is homeomorphic to
cw_D(R^n). In particular, we prove that the hyperspace cw(R^n) of all compact
convex bodies of constant width, as well as the hyperspace crw(R^n) of all
pairs of compact convex sets of constant relative positive width are
homeomorphic to R^{n+1}\times Q.
