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For every n ≥ 1, let cc(R n ) denote the hyperspace of all non-empty compact convex subsets of the Euclidean space R n endowed with the Hausdorff metric topology. For every non-empty convex subset D of [0, ∞) we denote by cw D (R n ) the subspace of cc(R n ) consisting of all compact convex sets of constant width d ∈ D and by crw D (R n ) the subspace of the product cc(R n ) × cc(R n ) consisting of all pairs of compact convex sets of constant relative width d ∈ D. In this paper we prove that cw D (R n ) and crw D (R n ) are homeomorphic to D × R n × Q, whenever D = {0} and n ≥ 2, where Q denotes the Hilbert cube. In particular, 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 × Q.2010 Mathematics Subject Classification. 57N20, 57S10, 52A99, 52A20, 54B20, 54C55.

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