# The Bishop--Phelps--Bollob\'as property for Lipschitz maps

## TL;DR

This paper introduces a Lipschitz version of the Bishop-Phelps-Bollobás property, establishing conditions under which Lipschitz maps can be approximated while strongly attaining their norm, with applications to various metric spaces.

## Contribution

It defines the Lip-BPB property for Lipschitz maps and proves it holds for finite pointed metric spaces, finite-dimensional Banach spaces, and certain classes of metric spaces like ultrametric and H"older spaces.

## Key findings

- Lip-BPB property holds for finite pointed metric spaces and finite-dimensional Banach spaces.
- It extends to uniformly Gromov concave, finite concave, ultrametric, and H"older metric spaces.
- The property can be extended from real-valued Lipschitz maps to maps into Banach spaces.

## Abstract

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollob\'as property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness assumptions are needed. Next, we show that if $M$ is a uniformly Gromov concave pointed metric space (i.e.\ the molecules of $M$ form a set of uniformly strongly exposed points), then $(M,Y)$ has the Lip-BPB property for every Banach space $Y$. We further prove that this is the case for finite concave metric spaces, ultrametric spaces, and H\"older metric spaces. The extension of the Lip-BPB property from $(M,\mathbb R)$ to some Banach spaces $Y$ and some results for compact Lipschitz maps are also discussed.