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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.
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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