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

TL;DR

This paper investigates small diameter properties of the unit ball in Banach spaces, introduces the Ball Huskable Property, compares it with related properties, and studies their stability under various space constructions.

Contribution

It introduces the Ball Huskable Property ($BHP$), compares it with related properties, and analyzes their stability under sums and three space property conditions.

Findings

$BDP$ implies $BHP$, which implies $BSCSP$; none of these implications are reversible.

All properties are stable under $l_p$ sums, $c_0$ sums, and Lebesgue Bochner spaces.

$BHP$ is a three space property when $X/Y$ is finite dimensional.

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}).