An Asymptotic Property of Schachermayer's Space under Renorming

Abstract

AbstractLet 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 ϵ>0 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−δ, respectively, (1/(k+1))||∑k+1i=1xni||≤1−δ. 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