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Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to $x_i$ than to any other point $x_j$. The partition of the whole space $\mathbb{R}^d$ by the Voronoi cells $C(x_i)$ is referred to as a Voronoi tessellation.

A natural extension of this construction involves replacing the points $x_{1},x_{2},\dots$ by mutually disjoint, bounded, convex sets $K_{1},K_{2},\dots\subset\mathbb{R}^{d}$ so that the (generalized) Voronoi cell of the set $K_{i}$, denoted by $C(K_{i})$, consists of all the points in $\mathbb{R}^{d}$ that are closer to $K_{i}$ than to any other set $K_{j}$. In the reference book Stochastic Geometry and its Applications, the partition of $\mathbb{R}^{d}$ by the cells $C(K_{i})$ is referred to as a Voronoi S-tessellation.

A lot is known about the statistics of random Voronoi tessellations when the underlying points $x_{i}$ are sampled from a Poisson point process. Statistical quantities such as the mean volume of a typical Voronoi cell have been explicitly computed as functions of the intensity $\lambda$ of the Poisson point process (see Chapter 9 of Stochastic Geometry and its Applications). In contrast, much less seems to be known about random Voronoi S-tessellations, in particular, when the sets $K_{i}$ are taken to be identical disks ($d=2$) or balls ($d=3$), as one would get by sampling from a Gibbs point process with hard-sphere interactions (again at some intensity $\lambda$). Indeed, in this case, online searches yield mostly simulation results and non-rigorous, heuristic arguments.

I am wondering if there is any known rigorous result on the statistics of such random Voronoi S-tessellations (e.g., the mean volume of a typical Voronoi cell) or any rigorous method that may be relevant, especially those applicable to large $\lambda\gg1$.

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  • $\begingroup$ I'm almost certainly missing something obvious, but isn't the Vornoi diagram of a collection of identical discs exactly the Vornoi diagram of their centers? The distance from any exterior point to a disc is the distance from that point to the center of the disc minus the disc's radius, so when trying to figure out the Vornoi cells all the points will have the same 'offset' applied to their distances and none of the comparisons will be changed. $\endgroup$ Commented Sep 9 at 22:36
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    $\begingroup$ @StevenStadnicki You're completely right that the Voronoi diagram of a set of identical disks coincides with the Voronoi diagram of their centers. However, the point is that when sampling a set of non-overlapping disks, their positions cannot be independently chosen as in the Poisson case: the presence of a disk at position $x$ forbids placing additional disks centered at a distance of $\le 2R$ from $x$, where $R$ is the radius of a single disk. This "interaction" is what makes everything hard (but also much more interesting). $\endgroup$
    – Qidong He
    Commented Sep 10 at 16:40

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