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.
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机译：我们证明，在具有负曲率奇异性的磁盘上，所有欧几里得，双曲锥度都适用于欧几里得平面和双曲平面中的等距不等式。这是Weil和Bol定理的离散模拟，这些定理处理的黎曼曲率量度分别由0和-1限制。 A.D. Alexandrov证明了更强的离散形式，随后通过近似扩展到有界积分曲率的度量。我们的证明使用了度量的“离散共形变形”，消除了奇异性并增加了面积。因此，它类似于使用统一定理和次谐波函数的谐波次要分量的Weil的论点。
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