The Bishop–Phelps–Bollobás property for Lipschitz maps. (November 2019)

• Record Type: Journal Article
• Authors: Chiclana, Rafael; Martín, Miguel
• Is Part Of: Nonlinear analysis. Volume 188(2019)
• Page Start: 158
• Page End: 178
• Publication Date: 2019-11
• Subjects: primary 46B04 -- secondary 46B20 26A16 54E50; Lipschitz function -- Lipschitz map -- Lipschitz-free space -- Norm attaining operators -- Bishop–Phelps–Bollobás property -- Metric space; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.06.002
• Languages: English
• ISSNs: 0362-546X

Abstract

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.