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"][1],  *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"][2] by Schneider). 
  


  [1]: https://doi.org/0.4310/jdg/1214428822
  [2]: https://doi.org/10.1007/BF02844505