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,R) to some Banach spaces Y and some results for compact Lipschitz maps are also discussed.