The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study

Consider a network linking the points of a rate-$1$ Poisson point process on the plane. Write $\Psi^{\mbox{ave}}(s)$ for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most $s$ times the Euclidean distance. We give upper and lower bounds on the function $\Psi^{\mbox{ave}}(s)$, and on the analogous "worst-case" function $\Psi^{\mbox{worst}}(s)$ where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent $\alpha$ such that each function has $\Psi(s) \asymp (s-1)^{-\alpha}$ as $s \downarrow 1$.