The validity of the ``lim inf'' formula and a characterization of Asplund spaces

We show that for a given homology beta on a Banach space X the following ``lim inf'' formula lim inf T beta (C; x') C T-c(C; x) x' -> x holds true for every closed set C C X and any x epsilon C, provided that the space X x X is beta-trusted. Here T-beta (C; x) and T-c(C; x) denote the beta-tangent cone and the Clarke tangent cone to C at x. The trustworthiness includes spaces with an equivalent beta-differentiable norm or more generally with a Lipschitz beta-differentiable bump function. As a consequence, we show that for the Frechet bornology, this ``lim inf'' formula characterizes in fact the Asplund property of X. We use our results to obtain new characterizations of T beta-pseudoconvexity of X. (C) 2014 Elsevier Inc. All rights reserved.