For any norm on with countably many extreme points, we prove that there is a set of Hausdorff dimension 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 . 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 boundary.
Cite this article
Christopher J. Bishop, Hindy Drillick, Dimitrios Ntalampekos, Falconer’s distance set conjecture can fail for strictly convex sets in . Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1953–1968DOI 10.4171/RMI/1254