Critical transmission ranges (or radii) in wireless ad-hoc and sensor networks have been extensively investigated for various performance metrics such as connectivity, coverage, power assignment and energy consumption. However, the regions on which the networks are distributed are typically either squares or disks in existing works, which seriously limits the usage in real-life applications. In this article, we consider a convex region (i.e., a generalisation of squares and disks) on which wireless nodes are uniformly distributed. We have investigated two types of critical transmission radii, defined in terms of k-connectivity and the minimum vertex degree, respectively, and have also established their precise asymptotic distributions. These make the previous results obtained under the circumstance of squares or disks special cases of this work. More importantly, our results reveal how the region shape impacts on the critical transmission ranges: it is the length of the boundary of the (fixed-area) region that completely determines the transmission ranges. Furthermore, by isodiametric inequality, the smallest critical transmission ranges are achieved when regions are disks only.

翻译：对无线临时热量和传感器网络的关键传输范围(或弧度)进行了广泛调查,以了解各种性能衡量标准,如连接、覆盖面、电力分配和能源消耗等,然而,网络分布的区域通常是现有工程中的方形或磁盘,严重限制了实际应用中的使用。在本篇文章中,我们考虑了无线节点统一分布的锥形区域(即方形和磁盘的概观),我们分别调查了K-连通性和最低顶点度的两种关键传输弧度,并确定了其精确的脉冲分布。这使得以前在平方形或磁盘情况下取得的结果成为这项工作的特殊案例。更重要的是,我们的结果揭示了该区域如何影响关键传输范围:完全决定传输范围的是(固定区域)区域边界的长度。此外,由于测量的不平等性,只有区域是磁盘,才达到最小的关键传输范围。