An Asymptotic Property of Schachermayer’s Space under Renorming

Let X be a Banach space with closed unit ball B. Given k ∈ , 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 k i=1, respectively, ni k+1 i=1 , such that 1/ k + 1 x + k i=1 xni ≤ 1 − δ, respectively, 1/ k + 1 k+1 i=1 xni ≤ 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.