*Accepted Paper*

**Inserted:** 8 nov 2019

**Last Updated:** 25 nov 2020

**Journal:** Comm. Contemp. Math.

**Year:** 2019

**Abstract:**

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.

**Keywords:**
Torsional rigidity, Convex domains, shape optimisation, principal eigenvalue

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