Minimizing capacity among linear images of rotationally invariant conductors

Abstract

Logarithmic capacity is shown to be minimal for a planar set having $N$-fold rotational symmetry ($N \geq 3$), among all conductors obtained from the set by area-preserving linear transformations. Newtonian and Riesz capacities obey a similar property in all dimensions, when suitably normalized linear transformations are applied to a set having irreducible symmetry group. A corollary is P\'{o}lya and Schiffer's lower bound on capacity in terms of moment of inertia.