Stability Results Of Small Diameter Properties In Banach Spaces

Abstract

The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related geometric properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrarily small diameter. We show $BDP$ implies $BHP$ which in turn implies $BSCSP$ and none of the implications can be reversed. We prove similar results for the $w^*$-versions. We prove that all these properties are stable under $l_p$ sum for $1\leq p \leq \infty$. These stability results lead to a discussion in the context of ideals of Banach spaces. We prove that $BSCSP$ (respectively $BHP$, $BDP$) can be lifted from an M-Ideal to the whole space. We also show similar results for strict ideals. We note that the space $C(K,X)^*$ has $w^*$-$BSCSP$ (respectively $w^*$-$BHP$, $w^*$-$BDP$) when K is dispersed and $X^*$has the $w^*$-$BSCSP$ (respectivley $w^*$-$BHP$, $w^*$-$BDP$).