An asymptotic property of Schachermayer's space under renorming

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A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every > 0, there exists δ> 0 such that for every -separated sequence (xₙ) in the unit ball B, and every x in B, there are distinct indices m and n such that ||x + xₘ + xₙ|| < 3(1 - δ), respectively, ||xₘ + xₙ|| < 2(1 - δ). It is shown that a Banach space constructed by Schachermayer has property 2-beta but cannot be renormed to have property 2-NUC.