The Bishop–Phelps–Bollobás property for Lipschitz maps. (November 2019) - Chiclana, Rafael; Martín, Miguel
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, R ) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.