Suppose that X is a real or complex Banach space with norm | • |. Then X is a Hilbert space if and only if for all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and \x -Y\ ^ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function r) : {(z,y) £ X X X : \x -y\ 2 } -> R such that 7/(0,0) = 2 and If the condition 17(0,0) -2 is

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... ted, then the existence of such a function 77 characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).