  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.
