It's a 2D problem: If you know the difference between circles circumscribed around vs circles inscribed in a polygon, that's probably all the geometry you need.

It's a 2D problem: If you know the difference between circles circumscribed around vs circles inscribed in a polygon, that's probably all the geometry you need. The proof is a series of inequalities about the various areas inside the circumscribed circle and outside the polygon, and also the parts of the circumscribed circle that would be cut off by a tangent to the inscribed circle at the points where it touches the polygon. Does that help a bit?