The Bishop--Phelps--Bollobás property for Lipschitz maps

Abstract

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás 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 Hölder 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.

arXiv commentRevised version, some parts have been removed

References in corpus (8)

• The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B
• On strongly norm attaining Lipschitz maps
• Norm attaining Lipschitz functionals
• Lipschitz-free spaces over ultrametric spaces
• On the preserved extremal structure of Lipschitz-free spaces
• The version for compact operators of Lindenstrauss properties A and B
• Ultrametric subsets with large Hausdorff dimension
• The Bishop-Phelps-Bollobás property and absolute sums

Cited by in corpus (3)

• Residuality in the set of norm attaining operators between Banach spaces
• Emerging notions of norm attainment for Lipschitz maps between Banach spaces
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces