Almost preserved extreme points

TL;DR

This paper introduces almost preserved extreme points (APEPs) as a relaxation of preserved extreme points, characterizing the Radon-Nikodým property in Banach spaces through APEPs and exploring their presence in various classical spaces.

Contribution

It defines APEPs, establishes their equivalence with the RNP in Banach spaces, and analyzes their occurrence in classical and tensor product spaces, addressing an open problem.

Findings

Banach space has RNP iff every bounded set has an APEP

Unit ball of any equivalent renorming has an APEP iff RNP holds

Partial solution to the open problem on preserved extreme points in tensor products

Abstract

In this paper we introduce the notion of an almost preserved extreme point (APEP) of a set as a weakening of the concept of preserved extreme points, and we systematically study such points. As a main result, we prove that a Banach space $X$ has the Radon-Nikod\'{y}m property (RNP) if and only if every closed, convex, and bounded subset of the space has an APEP. Similarly, we prove that $X$ has the RNP if and only if the unit ball of every equivalent renorming has an APEP. We further investigate APEPs of the unit ball of classical Banach spaces, absolute sums, Lipschitz-free spaces, and projective tensor products. In the latter setting, our work also describes the preserved extreme points in the unit ball under the assumption that every bounded operator is compact, thereby partially solving an open problem.