Regularity of distance measures and sets

Let mu be a Radon measure with compact support in R-n such that .integral integral \x - y\(-alpha) d mu x d mu y < infinity for some alpha, (n + 1)/2 less than or equal to alpha < n. We show that the image of mu x mu under the distance map (x, y) bar right a rrow \x - y\ is an absolutely continuous measure with density of class Calp ha-(n+1)/2. As a corollary we get that if A subset of R-n is a Suslin set w ith Hausdorff dimension greater than (n+1)/2, then the distance set {\x - y \ : x, y is not an element of A) has non - empty interior