Condenser capacity and hyperbolic diameter

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 $t$.

arXiv comment15 pages, 5 figures

References in corpus (5)

• Dimensions, Whitney covers, and tubular neighborhoods
• Fast solution of boundary integral equations with the generalized Neumann kernel
• Fast and accurate computation of the logarithmic capacity of compact sets
• Conformal modulus on domains with strong singularities and cusps
• The sharp quantitative isocapacitary inequality (the case of $p$-capacity)

Cited by in corpus (1)

• Möbius metric in sector domains