Alexandrov's theorem, defines the notion of twice differentiable(smooth) and shows that convex functions have this property, almost everywhere. See:
https://en.wikipedia.org/wiki/Alexandrov_theorem
Given a convex compact region $R$ in the plane, we can describe its lower and upper envelope by a convex and a concave function. See the figure, with $f_1$ as lower and $f_2$ as upper envelope.
In this way, we can define, for each point on the boundary of $R$, whether the point is smooth or not.
Now, if $R$ is rotated, we would get two potentially different functions. And potentially, there could be a point that was declared before as smooth and now after the rotation is not anymore.
I don't think this can happen, except for some potential special boundary cases.
My question: Is the smoothness of a convex object rotation invariant?