Fillmore showed that there are sets of constant width in $\mathbb R^d$ with analytic boundaries which have a trivial symmetry group (so these are very different from spheres; see "Symmetries of surfaces of constant width", J. Differential Geom., Vol 3, (1969), pp. 103-110).
Moreover, the set of bodies of constant width with analytic boundaries is dense in the space of all convex bodies of constant width in $\mathbb R^d$ with respect to the Hausdorff metric (see e.g. "Smooth approximation of convex bodies" by Schneider).
