We study the geometry of the scale-invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale-invariant Cassinian metric under Möbius transformations.

中文翻译：

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关于尺度不变的卡西尼度量的说明

我们研究了尺度不变的Cassinian度量的几何，并证明了当域是单位球或上半部空间时，该度量与双曲度量之间的尖锐比较不等式。我们还证明了在莫比乌斯（Möbius）变换下尺度不变的卡西尼度量的尖锐畸变不等式。