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arXiv · 1911.07221

The typical cell of a Voronoi tessellation on the sphere

Zakhar Kabluchko·Christoph Thaele
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Abstract

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 formulae 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.

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Zakhar Kabluchko, Christoph Thaele. 2019-11-17. The typical cell of a Voronoi tessellation on the sphere. https://arxiv.org/abs/1911.07221

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