On the relations between principal eigenvalue and torsional rigidity

We consider the problem of minimising or maximising 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 $H^1_0(Omega)$, while $T(Omega)$ is the torsional rigidity of $Omega$. The optimisation problem above is considered in the class of {it all domains} $Omega$, in the class of {it convex domains} $Omega$, and in the class of {it thin domains}. The full Blaschke-Santal'o diagram for $lambda(Omega)$ and $T(Omega)$ is obtained in dimension one, while for higher dimensions we provide some bounds.