Zhang, Zi Hou et al., Near convexity, near smoothness and approximative compactness of half spaces in Banach spaces, Acta mathematica Sinica. English series (Print) , 32(5), 2016, pp. 599-606
The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.