About the Blaschke-Santaló Diagram of Area, Perimeter and Moment of Inertia

We study the Blaschke-Santaló 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 axes of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by Pólya, in the simplified setting of double symmetry.