Blaschke-Santaló diagrams represent a powerful tool in Shape Optimization, to investigate the relations of triples of shape functionals, \[F_1, F_2, F_3\], defined on a suitable class of admissible shapes \[\mathcal A\]. The study consists in the determination of the range of the vector shape functional \[(F_1,F_2):\mathcal A\to \mathbb R^2\], under the constraint \[F_3=1\].
In this talk I will present two examples. The former deals with spectral type functionals (torsional rigidity, first Dirichlet eigenvalue, volume), the latter with purely geometric ones (area, perimeter, moment of inertia). The talk is based on works in collaboration with D. Zucco (Univ. Torino), A. Henrot (Univ. Lorraine), and R. Gastaldello (Mines-Nancy).