Small Combination of Slices, Dentability and Stability Results Of Small Diameter Properties In Banach Spaces

Abstract: In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in \cite{B3}, \cite{BR} ,\cite{EW}, \cite{GM} was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in \cite{GGMS}. We introduce the 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 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 $1\leq p \leq \infty, c_0$ sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that $BHP$ is a three space property provided $X/Y$ is finite dimensional and same is true for $BSCSP$ when $X$ has $BSCSP$ and $X/Y$ is strongly regular (\cite{GGMS}).