Isoperimetric properties of condenser capacity

Abstract

For compact subsets $E$ of the unit disk $\mathbb{D}$ we study the capacity of the condenser $\mathrm{cap}(\mathbb{D},E)$ 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.