  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, which uses the uniformization theorem and the harmonic minorant of a
subharmonic function.
