The typical cell of a Voronoi tessellation generated by \(n+1\) uniformly distributed random points on the d-dimensional unit sphere \(\mathbb {S}^d\) is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in \(\mathbb {R}^d\). Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases \(d\in \{2,3,4\}\) are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.

中文翻译：

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球体上 Voronoi 镶嵌的典型单元

研究了由\(n+1\) 个均匀分布的随机点在d维单位球体\(\mathbb {S}^d\)上生成的 Voronoi 镶嵌的典型单元。其˚F维矢量是与分配识别˚F通过产生一个测试”多面体维矢量Ñ在随机点 \（\ mathbb {R} ^ d \） 。为任何d提供了预期f向量的显式公式，并且分别研究了低维情况\(d\in \{2,3,4\}\)。这也意味着球面 Voronoi 镶嵌中k维面总数的明确公式。