Originally Posted by

**undefined** For Q2, the regular n-gon "feels" like it should be the right answer, but as for proof I'm also at a loss. Perhaps we can prove that the center of the circle must lie inside the polygon, then gain something by considering the triangles defined by the center and pairs of adjacent vertices of the polygon? (The sum of areas of these triangles equals the area of the polygon.)

Edit: Maybe something like: for a triple of adjacent vertices, the sum of areas of the two triangles is maximized when the middle vertex bisects the arc connecting the outer vertices. Thus any candidate polygon which has a triple of adjacent vertices such that the middle vertex does not bisect the associated arc can't be optimal. Thus the only possibility is the regular n-gon.