Condenser capacity and hyperbolic diameter

Abstract

<p>Given a compact connected set <em>E</em> in the unit disk B² , we give a new upper bound for the conformal capacity of the condenser (B², <em>E</em>) in terms of the hyperbolic diameter <em></em><em>t</em> of <em>E</em>. Moreover, for <em>t </em>>0, we construct a set of hyperbolic diameter <em>t</em> and apply novel numerical methods to show that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is called a Reuleaux triangle in hyperbolic geometry and it has constant hyperbolic width equal to <em>t</em>.<br></p>