Some Inequalities Involving Perimeter and Torsional Rigidity

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