Regularity of Distance Measures and Sets

Abstract

AbstractLet μ be a Radon measure with compact support in IRn such that equation image 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.