  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 $t$.
