The Gehring-Osgood characterization for uniform Euclidean domains asserts that these are precisely the domains in which quasihyperbolic distance is bounded above by a constant multiple of the so-called relative distance. We prove that a hyperbolic plane domain is uniform if and only if its hyperbolic distance is bounded above by a constant multiple of its relative distance. Similar results hold for uniform domains in the Riemann sphere, and also for Euclidean inner uniform domains.

中文翻译：

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均匀域和双曲距离

均匀欧几里得域的 Gehring-Osgood 表征断言，这些域正是准双曲距离以所谓的相对距离的常数倍为界的域。我们证明了双曲平面域是均匀的，当且仅当其双曲距离的边界为其相对距离的常数倍。类似的结果适用于黎曼球中的均匀域，也适用于欧几里德内部均匀域。