  Can the Minkowski sum of two compact convex bodies be made smoother by
rotating one of them? We construct two infinitely differentiable strictly
convex plane bodies such that after any generic rotation (in the Baire category
sense) of one of the summands the Minkowski sum is not five times
differentiable. On the other hand, if for one of the bodies the zero set of the
Gaussian curvature has countable spherical image, we show that any generic
rotation makes their Minkowski sum as smooth as the summands. We also improve
and clarify some previous results on smoothness of the Minkowski sum.
