Infinite-dimensional hyperspaces of convex bodies of constant width

There exists a natural embedding of the hyperspace of compact convex bodies of constant width in RnRn into those of Rn+1Rn+1. This allows us to consider the hyperspace of compact convex bodies of constant width in the space R∞=lim→Rn. We prove that this hyperspace is homeomorphic to Q∞=lim→Qn, where Q denotes the Hilbert cube.