Falconer’s $(K, d)$ distance set conjecture can fail for strictly convex sets $K$ in $\mathbb R^d$

Abstract

For any norm on $\mathbb{R}^d$ with countably many extreme points, we prove that there is a set $E \subset \mathbb{R}^d$ of Hausdorff dimension $d$ whose distance set with respect to this norm has zero linear measure. This was previously known only for norms associated to certain finite polygons in $\mathbb{R}^2$. Similar examples exist for norms that are very well approximated by polyhedral norms, including some examples where the unit ball is strictly convex and has $C^1$ boundary.