Ball separation characterization of dentability and related properties in Banach spaces

Abstract: In this article, we first analyze the dual connection between ball separation property and the three properties in Banach spaces : Ball Dentable Property (BDP), Ball Huskable Property (BHP) and Ball Small Comination of Slice Property (BSCSP). We also introduce some related concepts in point version, namely the notion of semi w^*-Point of Continuity and semi w^* -Small Combination of Slice point in B_{X^*} and establish their ball separation characterization. The characterization of semi w^*-PC plays a key role in establishing a new formulation of Property (II). We show that every point in S_{X^*} is semi w^*-PC of B_{X^*} if and only if X has property (II). At the end of this article, we introduce the notion of A-Small Combination of Slice point in B_{X^*} and obtain a necessary ball separation condition for its existence. This allows us to prove a more general version of the result given in [BB, Proposition 2.5].