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On Small Combination of Slices in Banach Spaces

The notion of Small Combination of Slices (SCS) in the unit ball of a Banach space was first introduced in [4] and subsequently analyzed in detail in [12] and [13]. In this work, we introduce the notion of BSCSP, which can be seen as a generalization of dentability in terms of SCS. We study certain stability results for the w∗-BSCSP leading to a discussion on BSCSP in the context of ideals of Banach spaces. We prove that the w∗-BSCSP can be lifted from a M -ideal to the whole Banach Space. We also prove similar results for strict ideals and U -subspaces of a Banach space. We note that the space C(K,X)∗ has w∗-BSCSP when K is dispersed and X∗ has the w∗-BSCSP.