Imagine a disc with $N$ radially displaceable masses $m_g$.
A total imbalance with respect to the center of the disc can be calculated as follows (using the respective radiuses $r_1,...,r_N$):
$$\vec{U}_s(r_1,...,r_N) = m_g \cdot \sum_{k=1}^N r_k \cdot e^{i \cdot 2\pi \cdot\frac{k - 1}{N}}$$
The radiuses are constrained by $$r_k \leq r_{max} \land \left(r_k \geq r_{min} \lor r_k = 0 \right)$$
Now the question is:
How can I determine, which $\vec{U}_s$ can be obtained by all combinations of $r_1,...,r_N$?
EDIT:
A brute-force approach (i.e. all combinations of the values ${0, 0.0065, 0.007125, 0.00775, 0.008375, 0.009}$)
for N=5 and some arbitrary values for $r_{min}$ and $r_{max}$ shows the following distribution:
Answer:
This is how it looks like, when it's working for N=7 (implemented with Mathatica)
http ://i.stack.imgur.com/krqf0.gif
RESULT
I managed to visualize them :)
Visualization of the Minkovski Sums (N=7) (done with Mathematica)
We have a Minkowski sum of $M\le N$ line segments with different orientations. (Actually, the fact that they have equal length, are aligned with the origin and at reguar angles does not really matter.)
The sum of two such segments is a parallelogram. The sum of three is an hexagon and so on. Depending on the $k$ for wich $r_k>0$, the polygons can be regular/centered or not.
Below a case with $N=7,M=4$.
You can compute the polygon as the convex hull of the vector sums of all segment endpoints (there are $2^M$ of them).
The points such that $r_k=0$ have no impact (except that if you compute the centroid, they modify the global scale).