Suppose I have two star shaped polygons in 2D. For simplicity, say the origin is located in the kernels of each.
I'm trying to prove any or all of these three conjectures:
The Minkowski sum of these two star shaped polygons is a star shaped polygon.
The origin is contained in this new star shaped polygon's kernel.
Two star shaped polygons' Minkowski sum is a star shaped polygon. It's kernel is the Minkowski sum of the two poylgons' kernels.
From some empirical testing with random polygons, #1 seems to be true, and I strongly suspect #2 and #3, but I can't find a good angle of attack to prove it. I've tried looking in some of the literature but I can't find anywhere any of these three conjectures are discussed.
I've found Polygonal Minkowski Sums via Convolution, which explores building Minkowski sums from the cycles in convolutions, and I think a constructive proof of #1 might be possible, but it's a bit beyond me.
Say you have two vectors $a\in A$ and $b\in B$ going from the origin to an arbitrary point in each of the input polygons. Then if 1 and 2 hold true, then for any such $a,b$ the sum $a+b$ must be within the new polygon. Well, as $A$ was star-shaped with the origin in the kernel, you have $\lambda a\in A$ for $\lambda\in[0,1]$. Likewise $\mu b\in B$ for $\mu\in[0,1]$. Thus you have $\lambda(a+b)=\lambda a+\lambda b\in A+B$. So yes, evry point of the Minkowski sum is visible from the origin, simply by moving at equal rate along two corresponding vectors in the input polygons.
Now for 3. If you have 1 and 2 established, you can build on that. Translating the input polygons will maintain these properties as long as the origin stays in the kernel of each. But the combined effect of two translations covering the kernels is a Minkowski sum of the kernels. So this gives you that the kernel of the Minkowski sum is at least the Minkowski sum of the kernels. To show that it can't be more, use a proof by contradiction. I'll leave the details to you for now.