In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.

中文翻译：

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黎曼流形上的泊松-沃罗尼瓦镶嵌

在本文中，我们考虑了黎曼流形$ M $和由固定点$ x_0 $和$ M $上强度$ \ lambda $的Poisson点过程的并集产生的Poisson-Voronoi镶嵌。当$ \ lambda $达到无穷大时，我们获得了与$ x_0 $相关的Voronoi单元的平均顶点数的两项渐近展开。估计的第一项等于欧几里得设置中顶点的平均数，而第二项涉及在$ x_0 $处的标量曲率$ M $。这用[19]中的以前的2D语句以适当而严格的框架来解决，并将其扩展到更高的维度。证明这一结果的关键工具是在黎曼设置中对Blaschke–Petkantschin类型的变量公式进行新的更改，从而得出流形的Ricci曲率。