Isoperimetric properties of condenser capacity

TL;DR

This paper investigates the isoperimetric properties of condenser capacity within the hyperbolic geometry of the unit disk, proposing that among hyperbolic triangles of equal area, the equilateral one minimizes capacity.

Contribution

It introduces a new approach using hyperbolic geometry set functionals to analyze condenser capacity and conjectures the minimal capacity property of equilateral hyperbolic triangles.

Findings

Experimental evidence supporting the conjecture.

Equilateral hyperbolic triangles likely minimize capacity among equal-area triangles.

New set functionals relate hyperbolic geometry to condenser capacity.

Abstract

For compact subsets $E$ of the unit disk $ \mathbb{D}$ we study the capacity of the condenser ${\rm cap}( \mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the least capacity.