Smoothness of Minkowski sum and generic rotations

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.