On relations between principal eigenvalue and torsional rigidity

We consider the problem of minimizing or maximizing the quantity lambda(Omega)T-q(Omega) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, lambda(Omega) denotes the first eigenvalue of the Dirichlet Laplacian on H-0(1)(Omega), while T(Omega) is the torsional rigidity of Omega. The optimization problem above is considered in the class of all domains Omega, in the class of convex domains Omega, and in the class of thin domains. The full Blaschke-Santalo diagram for lambda(Omega) and T(Omega) is obtained in dimension one, while for higher dimensions we provide some bounds.