Let $X$ be a Banach space. Let $C$ be a subset of $X$. Let $x^*$ be a functional on $X$. Then $S(C, x^*, \alpha) := \{ x \in C : x^*(x) > \sup x^*(C) - \alpha \}$, $\alpha > 0$, is called the open slice of $C$ determined by $x^*$ and $\alpha$. $X$ has Radon Nikodym Property if and only all closed bounded convex sets admit slices of arbitrarily small diameter i.e. these sets are dentable. The geometry of Banach space is an area of research which characterizes the topological and measure theoretic concepts in Banach spaces in terms of geometric structure of the space. The related concepts were initiated developed and extensively studied in the context of Radon Nikodym Property and Krein Milman Property by Ghoussoub, Godefroy, Maurey, and Scachermayer [Memoirs AMS 1987]. In this work, we look at Banach spaces where the unit ball admits slices of arbitrarily small diameter. We look at some related properties as well. We prove that all these properties are stable under $l_p$ sum for $1 \leq p \leq ∞$, sum and Lebesgue Bochner spaces. We show that these are three space properties under certain conditions on the quotient space. We also study these properties in ideals of Banach spaces. This is based on two papers jointly written with my graduate student, Susmita Seal in [J. Math. Anal. Appl. 2022] and [J. Convex Anal. 2023]. The only prerequisite for this talk is the statement of the Hahn Banach Theorem.

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Last updated: 08 Oct 2024