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^d with 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)/(2d-2). 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)/(2d-2) . If q≤0, then the product is minimised among all bounded sets by a ball of measure 1.