If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls? 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!
 A: This is almost certainly false.  The following animation shows two convex shapes (with outlines shown in red and green) whose Minkowski sum is a disk of radius 3 (with outline shown in blue).  The green shape is an ellipse with major and minor radii 1 and 1/2, which uniquely determines the red shape.

I do not have a proof that the red shape is convex, but it shouldn't be too hard to check.
Incidentally, here is the Mathematica code I used to produce this animation:
MyPlot = ParametricPlot[{3*{Cos[t], Sin[t]},
          With[{u = ArcTan[-Sin[t], Cos[t]/2]}, 
               3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}]},
               {t, 0, 2 Pi}];
myframes = 
  Table[With[{u = ArcTan[-Sin[t], Cos[t]/2]}, 
    With[{pt = 3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}}, 
     Show[MyPlot, 
      ParametricPlot[pt + {Cos[r], Sin[r]/2}, {r, 0, 2 Pi}, 
       PlotStyle -> Darker[Green]], 
      Graphics[{PointSize[Large], Point[pt]}]]]], {t, 0, 2 Pi - Pi/20,
     Pi/20}]; ListAnimate[myframes]
Edit: Here is a simpler solution using two congruent shapes.  The boundary of each shape is the union of two circular arcs, each of which is congruent to 1/4 of the blue circle.

A: 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.
A: Here's mine.  Done before I saw Jim's solution (honest).  But after seeing his, I animated mine, too (using Maple).  
Two copies of the Reuleaux triangle  
http://en.wikipedia.org/wiki/Reuleaux_triangle 
same size, one rotated by 180 degrees from the other.  

