Regularity of Distance Measures and Sets

Abstract

Let μ be a Radon measure with compact support in IRn such that We show that the imw of μ x μ under the distance map (x, y) |x- y| is an absolutely continuous measure with density of class Ca-(n+1)/2. As a corollary we get that If AC IRn is a Suslin set with Hausdorff dimension greater than (n+1)/2, then the distance set {|x-y| : x, y ϵ A} has non-empty interior.