On Inscribed Polygons

Basic Question: If two convex polygons P1 and P2 both consist of the same set of edges but in different order and are hinged at vertices and we deform both at their vertices to maximize the enclosed area, will both chains yield the same max enclosed area?

Answer: It is all on: http://www.drking.org.uk/hexagons/misc/polymax.html. There is a nice proof that the area of a polygon with hinged edges all of fixed length is a maximum if the polygon is inscribed in a circle. The number or even the order of edges does not matter. And yes, any closed and hinged polygonal chain can be deformed at vertices so that it becomes a cyclic polygon.

Further question:

Consider a 2D smooth convex figure C and a convex closed polygon P, both deformable as follows: The sides of P are hinged at its vertices and free to rotate but we require P to be planar and convex throughout any deformation at these hinges. On the other hand, only scaling is allowed for C.

Claim: Using the transormations specified above, we can always change P and C such that the final form of P can be inscribed in the final form of C ( ie all vertices of P lie on the boundary of C).

Remark: As of 13th April 2015, one doesnt know if the claim holds. It is likely to hold. Maybe any closed and hinged polygonal polygonal chain can be inscribed in the homothet of any smooth figure (not only smooth convex figures).