The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study - Aldous, David; Lando, Tamar (arXiv.org, 2014)
Consider a network linking the points of a rate-\(1\) Poisson point process on the plane. Write \(Ψ^{ave}(s)\) 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 \(Ψ^{ave}(s)\), and on the analogous "worst-case" function \(Ψ^{worst}(s)\) 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 \(α\) such that each function has \(Ψ(s) \asymp (s-1)^{-α}\) as \(s \downarrow 1\).