Asymptotics of power-weighted Euclidean functionals

Let {Xi: i⩾1} be i.i.d. points in Rd, d⩾2, and let LMM({X1,…,Xn},p), LMST({X1,…,Xn},p), LTSP({X1,…,Xn},p), be the length of the minimal matching, the minimal spanning tree, the traveling salesman problem, respectively, on {X1,…,Xn} with weight function w(e)=|e|p. If the common distribution satisfies certain regularity conditions, then the strong law of large numbers for the above three Euclidean functionals, 1⩽p<d, has been obtained. In this paper we show that the same type of result holds for 0<p<1.