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

• Christopher J. Bishop ^[1] ; Hindy Drillick ^[2] ; Dimitrios Ntalampekos ^[1]
  1. [1] Stony Brook University

     Stony Brook University

     Town of Brookhaven, Estados Unidos

     
  2. [2] Columbia University

     Columbia University

     Estados Unidos

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 37, Nº 5, 2021, págs. 1953-1968
• Idioma: inglés
• DOI: 10.4171/rmi/1254
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.