Nondentable Sets in Banach Spaces

Stephen J. Dilworth, Chris Gartland, Denka Kutzarova, N. Lovasoa Randrianarivony

Research output: Contribution to journalArticlepeer-review


In his study of the Radon-Nikodým property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set A that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain’s result: in any bounded, nondentable set A (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of A-valued quasimartingales with sharply divergent behavior.

Original languageEnglish (US)
Pages (from-to)31-40
Number of pages10
JournalJournal of Convex Analysis
Issue number1
StatePublished - 2021


  • Convex sets
  • Dentable sets in normed spaces
  • Extreme points
  • Martingale convergence
  • Radon-Nikodým property

ASJC Scopus subject areas

  • Analysis
  • General Mathematics


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