The hypertangent cone

In Borwein and Strojwas [5] we have observed that in Baire metrizable spaces the hypertangent cone plays an important role in the theory of tangent cones and generalized subgradients, because its properties relate to Lipschitz behaviour of sets and functions. Here, more accurate results for Banach spaces are presented. They follow from a formula which may be viewed as the discrete version of the Treiman inclusion [18]; we prove this in Section 1. Our discrete formula has many applications. It implies generalizations of the Bishop-Phelps theorem on the density of support points [ 11 and the dense Clarke subdifferentiability theorem of McLinden [14]. In Section 2 we present a new characterization of the Clarke derivative and new conditions characterizing directionally Lipschitzian functions on a Banach space. In Section 3 we use our discrete formula to obtain mean value theorems for lower semicontinuous functions on Banach spaces in the vein of Penot [16]. Also we prove a strengthening of Lebourg’s [13] mean value theorem for reflexive spaces. In Section 4 we prove a general inversion theorem. In particular we use it to furnish conditions for the Clarke tangent cone of the intersection of sets to contain the intersection of the tangent cones for these sets. This generalizes the finite dimensional version of Ioffe [ll] and an epi-Lipschitzian version of Rockafellar [16] in a Banach space. The basic facts about tangent cones and generalized subgradients that we use in this paper are discussed in Clarke [l0]