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, repectively, (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 2NUC 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. In [2], R. Huff introduced the notion of nearly uniform convexity (NUC). A Banach space X with closed unit ball B is said to be NUC if for any ε > 0, there exists δ < 1 such that for every ε-separated sequence in B, co((xn))∩δB 6= ∅. Here co(A) denotes the convex hull of a set A; a sequence (xn) is ε-separated if inf{‖xn − xm‖ : m 6= n} ≥ ε. Huff showed that a Banach space is NUC if and only if it is reflexive and has the uniform Kadec-Klee property (UKK). Recall that a Banach space X with closed unit ball B is said to be UKK if for any ε > 0, there exists δ < 1 such that for every ε-separated sequence (xn) in B which converges weakly to some x ∈ X, we have ‖x‖ ≤ δ. A recent result of H. Knaust, E. Odell, and Th. Schlumprecht [3] gives an isomorphic characterization of spaces having NUC. They showed that a separable reflexive Banach space X is isomorphic to a UKK space if and only if X has finite Szlenk index. Another property related to NUC is the property (β) introduced by Rolewicz [7]. In [4], the first author showed that a separable Banach space X has property (β) if and only if both X and X∗ are NUC. In [5], a sequence of properties lying in between (β) and NUC are defined. Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (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)i=1, respectively, (ni) k+1 i=1 , such that 1 k + 1 ‖x+ k ∑