About the Blaschke-Santalo diagram of area, perimeter and moment of inertia

We study the Blaschke-Santal'o diagram associated to the area, the perimeter, and the moment of inertia. We work in dimension 2, under two assumptions on the shapes: convexity and the presence of two orthogonal axis of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by P'olya, in the simplified setting of double symmetry.