A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions

Abstract

Let M be a metric space and X be a Banach space. In this paper we address several questions about the structure of F(M )̂ ⊗π X and Lip0(M, X). Our results are the following: (1) We prove that if M is a length metric space then Lip0(M, X) has the Daugavet property. As a consequence, if M is length we obtain that F(M )̂ ⊗π X has the Daugavet property. This gives an affirmative answer to [13, Question 1] (also asked in [24, Remark 3.8]). (2) We prove that if M is a non-uniformly discrete metric space or an unbounded metric space then the norm of F(M )̂ ⊗π X is octahe- dral, which solves [6, Question 3.2 (1)]. (3) We characterise all the Banach spaces X such that L(X, Y ) is octahedral for every Banach space Y , which solves a question by Johann Langemets.