Small Combination of Slices and Dentability in Ideals of Banach Spaces

We study small diameter properties of the closed unit ball namely Ball Small Combination of Slices Property ( BSCSP ), Ball Huskable property ( BHP ) and Ball Dentable Property ( BDP ) in a Banach space and its dual in the context of ideals in Banach spaces. The related concepts for an arbitrary closed bounded convex set in a Banach space were initiated and developed by several authors in the 1970s and 1980s. These concepts were extensively studied in the context of small combination of slices, dentability, huskability, Radon Nikodym Property and Krein Milman Property by N. Ghoussoub, G. Godefroy, B. Maurey and W. Schachermayer [ Some topological and geometrical structures in Banach spaces , Memoirs of the American Mathematical Society, Volume 70, Number 378 (1987)]. We show that if a Banach space X has BSCSP (respectively, BHP , BDP), then any M-ideal of X also has BSCSP (respectively, BHP , BDP). We further show that if Y is an M-ideal or a strict ideal of X, then weak star versions of these properties can be lifted from Y * to X *. We use these results to prove the stability of these properties in C (K) spaces and their dual. Lastly, we show that if Y is an almost isometric ideal of X, then BSCSP (respectively BHP , BDP) can be lifted from Y to X.