I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions.
The issue is that these solutions rely on picking an appropriate point $C$ in the interior of the polygon.
The problem, more clearly, is to find a point $C$ in the interior of a polygon, such that every half segment connecting that point and the vertices of the polygon never intersects the Boundary of the polygon.
Examples:
In the first picture the point is a perfect candidate, in the second, the point is not because the red edge passes through a section outside the control polygon.
For now we can assume such a point exists, although for some shapes such point will not exist and so a more complex curve is needed.