The Hilbert metric on convex subsets of \({\mathbb {R}}^n\) has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subsets of complex projective spaces (see also L. Dubois in J Lond Math Soc (2) 79(3):719–737, 2009) and give examples of geometric applications. Basic examples include the hyperbolic metric on complex hyperbolic spaces, the n-punctured spheres and \(\mathbb {RP}^1\) minus a Cantor set.

中文翻译：

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无凸性的希尔伯特几何

\（{\ mathbb {R}} ^ n \）的凸子集上的希尔伯特度量已被证明是一个丰富的概念，并且已被广泛研究。在这里，我们建议将此度量推广到复杂射影空间的子集（另请参见L. Dubois在J Lond Math Soc（2）79（3）：719-737，2009）中给出了几何应用的示例。基本示例包括复杂双曲空间上的双曲度量，n个穿孔球体和\（\ mathbb {RP} ^ 1 \）减去Cantor集。