TITLE: Expected length of the shortest polygonal path connecting random points
QUESTION [30 upvotes]: $N$ points are selected in a uniformly distributed random way in a disk of a unit radius. Let $L(N)$ denote the expected length of the shortest polygonal path that visits each of the points at least once (the path need not to be closed any may be self-intersecting).

For what $N$ do we know the exact value of $L(N)$?
Is there a general formula for $L(N)$?
What is the asymptotic behavior of $L(N)$ as $N\to\infty$?
What are the answers to previous questions, if the disk is replaced with a ball?

REPLY [2 votes]: $L(2)=128/45\pi$
L(3) may or may not be extractable from here.
I suspect thats all thats known about exact values.
Also, your problem is called Traveling Salesman Problem.