Some Inequalities Involving Perimeter and Torsional Rigidity

We consider shape functionals of the form Fq(Ω) = P(Ω) Tq(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 Fq(Ω) is considered on various classes of admissible domains Ω : in the class Aall of all domains, in the class Aconvex of convex domains, and in the class Athin of thin domains.