# Strongly extreme points and approximation properties

## TL;DR

This paper investigates the relationship between strongly extreme points and denting points in Banach spaces, establishing conditions under which strongly extreme points are denting points and providing counterexamples.

## Contribution

It introduces conditions under which strongly extreme points are denting points and constructs a non-symmetric norm on c0 where all points are strongly extreme but not denting.

## Key findings

- Strongly extreme points are denting under certain approximation conditions.
- Banach spaces with the unconditional compact approximation property satisfy these conditions.
- A non-symmetric norm on c0 shows all points are strongly extreme but not denting.

## Abstract

We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting.