  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.
