Given n uniformly and independently distributed points in a ball of unit volume in dimension d, it is well established that the length of several combinatorial optimization problems (including the minimum spanning tree (MST), the minimum matching (M), the travelling salesman problem (TSP), etc.) on these n points is asymptotic to 
, where the constant 
depends on the dimension d and the problem solved. It has been a long open problem to determine the constants 
for these problems. In this paper progress is made in establishing the constants 
, 
for the MST and the matching problem. By applying Crofton's method, an old method in geometrical probability, it is proved that
, 
as d tends to infinity. Moreover, the method presented here corresponds to heuristics for these problems, which are asymptotically exact as the dimension increases. Finally, the asymptotics for the TSP constant are examined.
