Falconer’s distance set conjecture can fail for strictly convex sets in

  • Christopher J. Bishop

    Stony Brook University, USA
  • Hindy Drillick

    Columbia University, New York, USA
  • Dimitrios Ntalampekos

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

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Abstract

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. 1953–1968

DOI 10.4171/RMI/1254