Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(Ω)=P(Ω)T^q(Ω)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(Ω)$ denotes the perimeter of $Ω$ and $T(Ω)$ is the torsional rigidity of $Ω$. The minimization and maximization of $F_q(Ω)$ is considered on various classes of admissible domains $Ω$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.

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