TITLE: If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?
QUESTION [17 upvotes]: If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't found a counterexample. I don't have experience with Minkowski sums so any help will be appreciated.
Thanks!

REPLY [2 votes]: For $\mu$ a Borel positive measure on the sphere $S^{n-1}$, consider a continuous Minkowski sum of segments
$$ K_\mu = \int_{S^{n-1}} [0,\theta] \, \mathrm{d}\mu (\theta) .$$
The set $K_\mu$ is convex (it could be defined by its support function, then the integral becomes a usual one). Now observe that (1) $K_{\mu+\nu} = K_\mu + K_\nu$ (2) by rotation invariance, $K_\mu$ is a Euclidean ball if $\mu$ is the uniform measure (3) there are many ways to write the uniform measure as a sum of positive measures.