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 ${\mathrm{\Psi }}^{ave}(s)$ 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 ${\mathrm{\Psi }}^{ave}(s)$, and on the analogous “worst-case” function ${\mathrm{\Psi }}^{worst}(s)$ 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 $\mathrm{\Psi }(s)\asymp (s-1{)}^{-\alpha }$ as $s\downarrow 1$.