关键词: weak$^*$-weak denting point, nearly weak dentability, very smooth space, point of weak$^*$-weak continuity, extreme point, approximatively weak compactness, $w$-strong proximinality, reflexivity

Abstract: In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak$^*$-weak denting point and the weak$^*$-weak$^*$ denting point of a set. These are the generalizations of the weak$^*$ denting point of a set in a dual Banach space. By use of the weak$^*$-weak denting point, we characterize the very smooth space, the point of weak$^*$-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the $w$-strong proximinality of every closed convex subset of Banach spaces.

Key words: weak$^*$-weak denting point, nearly weak dentability, very smooth space, point of weak$^*$-weak continuity, extreme point, approximatively weak compactness, $w$-strong proximinality, reflexivity