This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in ${\mathbb{R}}^d$, when restricted to the corresponding Voronoi diagram imposed on S. The diagram obtained by restricting the SINR zones to their corresponding Voronoi cells is referred to hereafter as an SINR+Voronoi diagram.

Uniform SINR diagrams, where all stations transmit with the same power, are simple and nicely structured, e.g., the station reception zones are convex and “fat”. In contrast, nonuniform SINR diagrams might be complex; the reception zones might be fractured and their boundaries might contain many singular points. In this paper, we establish the perhaps surprising fact that a nonuniform SINR+Voronoi diagram is topologically almost as nice as a uniform SINR diagram. In particular, it is convex and effectively⁵ fat. This holds for every power assignment, every path-loss parameter α and every dimension $d\ge 1$. The convexity property also holds for every SINR threshold $\beta >0$, and the effective fatness property holds for any $\beta >1$. These fundamental properties provide a theoretical justification to engineering practices basing zonal tessellations on the Voronoi diagram, and help to explain the soundness and efficacy of such practices.

We also consider two algorithmic applications. The first concerns the Power Control with Voronoi Diagram (PCVD) problem, where given n stations embedded in some polygon $\mathcal{P}$, it is required to find the power assignment that optimizes the SINR threshold of the transmission station $s_i$ for any given reception point $p\in \mathcal{P}$ in its Voronoi cell

本文关注的行为SINR图无线系统，一组构成š的Ñ嵌入在站${\mathbb{电阻}}^d$，当限制为施加在S上的相应Voronoi 图时。通过将 SINR 区域限制到它们相应的 Voronoi 单元而获得的图在下文中称为SINR + Voronoi 图。

统一 SINR 图，其中所有站都以相同的功率发射，简单且结构良好，例如，站接收区是凸的和“胖”的。相比之下，非均匀 SINR 图可能很复杂；接收区可能会断裂，它们的边界可能包含许多奇异点。在本文中，我们建立了一个可能令人惊讶的事实，即非均匀 SINR+Voronoi 图在拓扑上几乎与均匀 SINR 图一样好。特别是，它是凸的，有效地⁵脂肪。这适用于每个功率分配、每个路径损耗参数α和每个维度$d\ge 1$. 凸性属性也适用于每个 SINR 阈值$\beta >0$，并且有效的胖度属性对任何 $\beta >1$. 这些基本属性为基于 Voronoi 图的分区镶嵌的工程实践提供了理论依据，并有助于解释此类实践的合理性和有效性。

我们还考虑了两种算法应用。第一个涉及Voronoi 图(PCVD) 问题的功率控制，其中给定嵌入某些多边形的n个站$\mathcal{磷}$, 需要找到优化传输站 SINR 阈值的功率分配 ${秒}_{\mathrm{一世}}$ 对于任何给定的接收点 $磷\in \mathcal{磷}$ 在它的 Voronoi 细胞中