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

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