# Emerging notions of norm attainment for Lipschitz maps between Banach spaces

## TL;DR

This paper classifies various notions of norm attainment for Lipschitz maps between Banach spaces, explores their relationships, and introduces a new property extending previous scalar-valued results, with implications for the density of such maps.

## Contribution

It provides a comprehensive classification of norm attainment notions for Lipschitz maps, analyzes their density properties, and introduces a new local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps.

## Key findings

- Characterization of Banach spaces with the Radon-Nikodým property via density of Lipschitz maps.
- Relations among different notions of norm attainment for Lipschitz maps.
- Introduction of the local directional Bishop-Phelps-Bollobás property for Lipschitz compact maps.

## Abstract

We classify several notions of norm attaining Lipschitz maps which were introduced previously, and present the relations among them in order to verify proper inclusions. We also analyze some results for the sets of Lipschitz maps satisfying each of these properties to be dense or not in $\Lip(X,Y)$. For instance, we characterize a Banach space $Y$ with the Radon-Nikod\'ym property in terms of the denseness of norm attaining Lipschitz maps with values in $Y$. Further, we introduce a property called the local directional Bishop-Phelps-Bollob\'as property for Lipschitz compact maps, which extends the one studied previously for scalar-valued functions, and provide some new positive results.