Suppose that one selects two random points x,y in a sphere of radius R. Is there a closed-form expression for the probability density P(d_x,y), i.e. the probability that x and y have Euclidean distance d in the sphere? Note that I'm referring to points within a sphere, not the probability density over the surface area.
Where $d$ is the distance and $R$ is the radius.
See for instance Tu and Fischbach, 2011 who also provide results for this problem in spherical objects of general dimension and density.