Given two closed curves, when is their minkowski sum differentiable?

Question

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve differentiable, and is there some relation between the differentiability of $\gamma_1$ and $\gamma_2$, and the differentiability of the resulting curve?

Answer 1

If you take the Minkowski sum of two half disks such that their union is a disk you will get a differentiable boundary, even though the boundaries of the two summands are not everywhere differentiable. So it's hard to predict when the Minkowski sum will have differentiable boundary. I believe it is true that if your two convex figures have differentiable boundary, then the Minkowski sum will also have differentiable boundary.