On relations between principal eigenvalue and torsional rigidity

We consider the problem of minimising or maximising the quantity λ(Ω)Tq(Ω) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, λ(Ω) denotes the first eigenvalue of the Dirichlet Laplacian on H10(Ω), while T(Ω) is the torsional rigidity of Ω. The optimisation problem above is considered in the class of all domains Ω, in the class of convex domains Ω, and in the class of thin domains. The full Blaschke-Santal´o diagram for λ(Ω) and T(Ω) is obtained in dimension one, while for higher dimensions we provide some bounds.