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Diameter 2 properties and convexity

Volume 232 / 2016

Trond Arnold Abrahamsen, Petr Hájek, Olav Nygaard, Jarno Talponen, Stanimir Troyanski Studia Mathematica 232 (2016), 227-242 MSC: Primary 46B04, 46B20. DOI: 10.4064/sm8317-4-2016 Published online: 25 April 2016


We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of $C[0,1]$ with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.


  • Trond Arnold AbrahamsenDepartment of Mathematics
    University of Agder
    Postbox 422
    4604 Kristiansand, Norway
  • Petr HájekMathematical Institute
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic
    Department of Mathematics
    Faculty of Electrical Engineering
    Czech Technical University in Prague
    Zikova 4
    160 00 Praha, Czech Republic
  • Olav NygaardDepartment of Mathematics
    University of Agder
    Postbox 422, 4604 Kristiansand, Norway
  • Jarno TalponenDepartment of Physics and Mathematics
    University of Eastern Finland
    Box 111
    FI-80101 Joensuu, Finland
  • Stanimir TroyanskiDepartamento de Matemáticas
    Universidad de Murcia
    Campus de Espinardo
    30100 Espinardo (Murcia), Spain
    Institute of Mathematics and Informatics
    Bulgarian Academy of Sciences
    bl. 8, acad. G. Bonchev St.
    1113 Sofia, Bulgaria

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