Every non-reflexive Banach space admits an equivalent norm with a unit sphere point that is strongly exposed by all supporting functionals but not almost locally uniformly rotund, while the 2004 characterization holds in reflexive spaces.
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Some remarks on almost locally uniformly rotund points
Every non-reflexive Banach space admits an equivalent norm with a unit sphere point that is strongly exposed by all supporting functionals but not almost locally uniformly rotund, while the 2004 characterization holds in reflexive spaces.