A simple proof of an isoperimetric inequality for Euclidean and hyperbolic cone-surfaces

We prove that the isoperimetric inequalities in the Euclidean and hyperbolic plane hold for all Euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by −1. A stronger discrete version was proved by A.D. Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature.Our proof uses “discrete conformal deformations” of the metric that eliminate the singularities and increase the area. Therefore it resembles Weil's argument that uses the uniformization theorem and the harmonic minorant of a subharmonic function.