Minimizing capacity among linear images of rotationally invariant conductors

TL;DR

This paper proves that among all area-preserving linear transformations, a planar set with N-fold rotational symmetry minimizes logarithmic capacity, extending similar results to Newtonian and Riesz capacities in higher dimensions.

Contribution

It establishes minimal capacity properties for symmetric sets under linear transformations, generalizing classical bounds and connecting capacity with symmetry and inertia.

Findings

Logarithmic capacity is minimized for N-fold symmetric sets under area-preserving linear maps.

Similar minimality properties hold for Newtonian and Riesz capacities in higher dimensions.

A corollary provides a lower bound on capacity based on the set’s moment of inertia.

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.