On capacity and torsional rigidity

We investigate extremality properties of shape functionals which are products of Newtonian capacity $\text{cp}(\overline{\text{Om}})$, and powers of the torsional rigidity $T(\text{Om})$, for an open set $\text{Om}\subset {\mathbb{R}}^d$ with compact closure $\overline{\text{Om}}$, and prescribed Lebesgue measure. It is shown that if $\text{Om}$ is convex then $\text{cp}(\overline{\text{Om}})T^q(\text{Om})$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$\mathrm{.}If$q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.