We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset { \Bbb R}^d$, $d \ge 2$, is greater than $\frac{ d+1}{ 2}$, then the distance set $\Delta(E)=\{ |x-y|: x,y \in E \}$ contains an interval. We prove this result for distance sets $\Delta_B(E)=\{ { ||x-y||}_B: x,y \in E \}$, where ${ || \cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.
Categories: math.CA math.MG