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A basis {xr' of a Banach space X is said to satisfy property A if for every number C>O there exists a number r,>O such that !!I>1 oc,xill=1 and jjE.+ n-cO tx!!OC imply oc, 1?x,x1 >I+r,. It is knowrn that property A implies: (1) {x?' is a boundedly complete basis of X, and (2) every convex, weakly compact subset of X has normal structure. In this paper, we construct a reflexive Banach space X, with an unconditional basis {x,}, such that: (a) X has normal structure, and (b) there does not exist an equivalent norm on X with respect to which {x,} satisfies property A; showing that the converse of (1) and (2) is invalid even with the weaker conclusion that {xn} be equivalent to a basis satisfying property A.

On a geometric condition related to boundedly complete bases and normal structure in Banach spaces