This is an announcement for the paper “The Bishop--Phelps--Bollobás property for Lipschitz maps” by Rafael Chiclanahttps://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R, Miguel Martinhttps://arxiv.org/search/math?searchtype=author&query=Martin%2C+M. 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 to make 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 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 of 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 $Y$, the relationship with absolute sums, and some results only valid for compact Lipschitz maps, are also discussed.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1901.02956