difficult to construct a compact set $E \subset {\mathbb{R}}^d$ with Hausdorff dimension $d$ so that $\Delta_{K}(E)$ has measure zero. But there are non-trivial results if $K$ is curved. We focus on the case that $\partial K$ is $C^\infty$ smooth and has positive Gaussian curvature. It is plausible that Falconer’s conjecture remains true for all such norms, and most previous results on the problem extend to this setting. For instance, Erdoğan’s bound extends to this class of norms – cf. Remark 1.6 in [@Erd05]. Our method also extends to this class of norms.

\[main2\] Let $K$ be a symmetric convex body in ${\Bbb R}^2$ whose boundary $\partial K$ is $C^\infty$ smooth and has strictly positive curvature. Let $E\subset\mathbb{R}^2$ be a compact set whose Hausdorff dimension is larger than $\frac54$. Then, there exists a point $x\in E$ so that the pinned distance set $$\Delta_{K,x}(E):=\{{||x-y||}_K:\, y\in E\}$$ has positive Lebesgue measure.

\[ksanalogrmk\] One can adapt the proof of Theorem \[main1\] and Theorem \[main2\] to yield the following result. Suppose that the Hausdorff dimension of a compact set $E \subset {\Bbb R}^2$ is equal to $s>1$ and $K$ is as in Theorem \[main2\]. Then there exists $x \in E$ such that the upper Minkowski 