We derive an asymptotic theorem generalizing in a sense the one by Beardwood et al. and Steele for the Traveling Salesman Problem, the minimun weight perfect matching problem and the minimum spanning tree problem on a random set of points in R^d, where the edge weight are taken to be equal to some power of the Euclidean distance. The rate of convergence was estimated by numerical experiments. These tend to show that for Euclidean distances, the length of the minimum spanning tree is asymptotically equal to slightly more than twice the weight of the optimum minimum perfect matching, while there seems to be a larger gap between the former and the length of the optimum traveling salesman tour.