N points are selected uniformly at random in the unit square. Let L(N) be the expected length of the shortest (possibly self-intersecting) [polygonal chain](http://en.wikipedia.org/wiki/Polygonal_chain) connecting all the points. It can be proved that $L(N)\sim c\sqrt{AN}$, for some constant $c$, where A is the area of the region (in our case A=1).
Heuristics suggests c around 0.7, is there known an exact value for the constant c?


The question including some others was [asked originally](https://math.stackexchange.com/questions/569534/expected-length-of-the-shortest-polygonal-path-connecting-random-points) on math.SE, where some partial results where achieved.


If instead of a polygonal chain, we use the minimal spanning tree, a similar argument yields the same asymptotic formula, is the constant c known for this version?



Expected length of the shortest polygonal chain connecting N random points in the unit square
Some partial answers:


1) The fact that the length is asymptotic to $c\sqrt{N}$ follows from sub-additivity.


2) This is the ``random travelling salesman'' problem. Joe Yukich wrote several papers on it and its variants. 


3) See also the wikipedia page
<http://en.wikipedia.org/wiki/Travelling_salesman_problem#TSP_path_length_for_random_pointset_in_a_square>


I do not think that the value of $c$ is known explicitly.


