The validity of the “liminf” formula and a characterization of Asplund spaces

We show that for a given bornology β on a Banach space X the following “liminf” formulaliminfx′⟶CxTβ(C;x′)⊂Tc(C;x) holds true for every closed set C⊂X and any x∈C, provided that the space X×X is ∂β-trusted. Here Tβ(C;x) and Tc(C;x) denote the β-tangent cone and the Clarke tangent cone to C at x. The trustworthiness includes spaces with an equivalent β-differentiable norm or more generally with a Lipschitz β-differentiable bump function. As a consequence, we show that for the Fréchet bornology, this “liminf” formula characterizes in fact the Asplund property of X. We use our results to obtain new characterizations of Tβ-pseudoconvexity of X.