TITLE: Distribution of pairwise distances
QUESTION [12 upvotes]: I am seeking results that describe the distribution of the set of
Euclidean distances between pairs of $n$ points in
a unit square in the plane.
For example: All the distances could be short (a tight cluster),
but not all the distances could be long (all long distances
force some short distances).
Are there results of this general form,
constraining the possible shapes of the histogram of 
distances, for large $n$?

          

 


          

Distribution of point-to-point distances in a unit square.


My actual application requires some constraints on the
lengths of geodesics between points sprinkled
on a surface embedded in $\mathbb{R}^3$,
a surface homeomorphic to a sphere.
I'm hoping the planar case will shed light on geodesic constraints.

REPLY [3 votes]: Let A be the measure on R^2 invariant wrt rotations about the origin with total weight at each radius equal to the frequency of that distance in your histogram. This measure is obtained by rotationally averaging the autocorrelation measure. The Fourier transform of the autocorrelation measure is nonnegative. Since the Fourier transform is equivariant under rotation, I believe this means that the Fourier transform of A must also be nonnegative. This yields some strong constraints on the histogram of distances, which are basically the entire theory behind LP bounds.