A characterization of reflexivity (Q676752)

Let \((X,|\cdot|)\) be a real Banach space with norm derivative \[ (x,y)_i= \lim_{t\to 0_-} \Biggl[{|y+ tx|^2-|y|^2\over 2t}\Biggr]. \] The author has proved that a Banach space \(X\) is reflexive if and only if for every continuous convex real-valued mapping \(F\) on \(X\) and any \(x_0\in X\) there exists a \(u_{F,x_0}\in X\) so that \(F(x)\geq F(x_0)+ (x- x_0,u_{F,x_0})_i\) for all \(x\in X\).