Condenser capacity and hyperbolic diameter

TL;DR

This paper establishes a new upper bound for the conformal capacity of condensers in the unit disk based on hyperbolic diameter and introduces a hyperbolic Reuleaux triangle with maximal capacity among sets of equal diameter.

Contribution

It provides a novel upper bound for condenser capacity in terms of hyperbolic diameter and constructs a hyperbolic Reuleaux triangle demonstrating maximal capacity for given diameter.

Findings

New upper bound for conformal capacity in terms of hyperbolic diameter

Construction of a hyperbolic Reuleaux triangle with maximal capacity

Numerical methods showing the Reuleaux triangle exceeds hyperbolic disk capacity

Abstract

Given a compact connected set $E$ in the unit disk $\mathbb{B}^{2}$, we give a new upper bound for the conformal capacity of the condenser $(\mathbb{B}^{2}, E)$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$, we construct a set of hyperbolic diameter $t$ 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 $t$.