An asymptotic property of Schachermayer's space under renorming

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 (x_n)⊆B there are indices (n_i)^k_i=1, respectively, (n_i)^k+1_i=1, such that (1/(k+1))||x+∑^k_i=1x_ni||<1-, respectively, (1/(k+1))||∑^k+1_i=1x_ni||<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.