Johan Philip derives the PDF for the distance between 2 random points in a unit square: http://www.math.kth.se/~johanph/habc.pdf
What is the PDF for the minimum separation of 3 random points in a unit square? (where the maximum minimum separation is known to be sqrt(6)-sqrt(2): http://en.wikipedia.org/wiki/Circle_packing_in_a_square
Similar question for k points where 3 <= k <= 9 (for which the maximum minimum separation is known precisely). Similar question for unit circle.