# On some local Bishop-Phelps-Bollob\'as properties

## TL;DR

This paper investigates specific local Bishop-Phelps-Bollobás properties for bounded linear operators, analyzing their relationships with geometric space properties and providing a comparative diagram of these properties.

## Contribution

It introduces and studies two new local Bishop-Phelps-Bollobás properties, L_{p, o} and L_{o, p}, and explores their connections with geometric properties of Banach spaces.

## Key findings

- Defined and analyzed the properties L_{p, o} and L_{o, p}
- Established relations between these properties and space geometries like strict convexity and local uniform rotundity
- Provided a diagram comparing all known Bishop-Phelps-Bollobás type properties

## Abstract

We continue a line of study about some local versions of Bishop-Phelps-Bollob\'as type properties for bounded linear operators. We introduce and focus our attention on two of these local properties, which we call L$_{p, o}$ and L$_{o, p}$, and we explore the relation between them and some geometric properties of the underlying spaces, such as spaces having strict convexity, local uniform rotundity, and property $\beta$ of Lindenstrauss. At the end of the paper, we present a diagram comparing all the existing Bishop-Phelps-Bollob\'as type properties with each other. Some open questions are left throughout the article.