An asymptotic property of Schachermayer's space under renorming

Denka Kutzarova, Denny H. Leung

Research output: Contribution to journalArticlepeer-review


Let X be a Banach space with closed unit ball B. Given k∈N, X is said to be k-β, respectively, (k+1)-nearly uniformly convex ((k+1)-NUC), if for every ε0 there exists δ, 0<δ<1, so that for every x∈B and every ε-separated sequence (xn)⊆B there are indices (ni)ki=1, respectively, (ni)k+1i=1, such that (1/(k+1))||x+∑ki=1xni||<1-, respectively, (1/(k+1))||∑k+1i=1xni||<1-;. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.

Original languageEnglish (US)
Pages (from-to)670-680
Number of pages11
JournalJournal of Mathematical Analysis and Applications
Issue number2
StatePublished - Oct 15 2000
Externally publishedYes


  • Nearly uniform convexity
  • Renorming
  • Schachermayer's space

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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