Sala Seminari (Dip. Matematica).
A Blaschke-Santalo’ diagram is the range of a vector shape functional $(F_1,F_2)$ in $\mathbb R^2$. The determination of such attainable set amounts to completely characterize the relation between $F_1$ and $F_2$. In this talk I will present some recent results obtained in collaboration with D. Zucco, in the case of $F_1$ the first Dirichlet eigenvalue and $F_2$ the inverse of the torsional rigidity, defined on convex shapes with unit volume, and, as a variant, on convex sets with volume at most 1.The study led us to address some very deep questions, whose answers are still open problems: in the last part of the talk, I will list them, together with our conjectures.