On capacity and torsional rigidity

We investigate extremal properties of shape functionals which are products of Newtonian capacity cap ( Ω ¯ ) , and powers of the torsional rigidity T ( Ω ) , for an open set Ω ⊂ R dwith compact closure Ω ¯ , and prescribed Lebesgue measure. It is shown that if Ω is convex, then cap( Ω ¯ ) T q ( Ω )is (i) bounded from above if and only if q ⩾ 1 , and (ii) bounded from below and away from 0 if and only if q ⩽ d − 2 2 ( d − 1 ). Moreover a convex maximiser for the product exists if either q > 1 , or d = 3 and q = 1 . A convex minimiser exists for q < d − 2 2 ( d − 1 ). If q ⩽ 0 , then the product is minimised among all bounded sets by a ball of measure 1.