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.

We consider the torsional rigidity and the principal eigenvalue related to the p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $$p=1$$ p = 1 and $$p=\infty $$ p = ∞ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.

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 A all of all domains, in the class A convex of convex domains, and in the class A thin of thin domains.

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $$P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}$$ P ( Ω ) T q ( Ω ) | Ω | - 2 q - 1 / 2 , and the class of admissible domains consists of two-dimensional open sets $$\Omega $$ Ω satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when $$q<1/2$$ q < 1 / 2 an optimal relaxed domain exists. When $$q>1/2$$ q > 1 / 2 , the problem is ill-posed, and for $$q=1/2$$ q = 1 / 2 , the explicit value of the infimum is provided in the cases $$k=0$$ k = 0 and $$k=1$$ k = 1 .

We study a general version of the Cheeger inequality by considering the shape functional F p,q (Ω) = λ 1/p p (Ω)/λ q (Ω) 1/q . The infimum and the supremum of F p,q are studied in the class of all domains Ω of R d and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for F p,q is discussed.