Falconer's (K, d) distance set conjecture can fail for strictly convex sets K in Rd

For any norm on Rd with countably many extreme points, we prove that there is a set E ⊂ Rd 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 R2. 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 C1 boundary.