An asymptotic property of Schachermayer's space under renorming

[摘要] Let X be a Banach space with closed unit ball B. Given k epsilon N, X is said to be k-beta, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every epsilon > 0 there exists delta, 0 < < 1, so that for every x B and every E-separated sequence (x(n)) subset of or equal to B there are indices (n(i))(i=1)(k), respectively, (n(i))(i-1)(k+1), such that (1/(k + 1))\\x + Sigma (k)(i=1) \\ less than or equal to 1 delta, respectively, (1/(k + 1))\\ Sigma (k+1)(i=1) x(ni) \\ 1- delta. It is shown that a Banach space constructed by Schachermayer is 2-beta, 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-beta. (C) 2000 Academic Press.

[发布日期] 2000-10-15

[关键词] nearly uniform convexity;renorming;Schachermayer's space