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

Let M be a metric space and X be a Banach space. In this paper we address several questions about the structure of $F\left(M\right){\hat{\otimes }}_{\pi }X$ and ${\mathrm{Lip}}_0(M,X)$. Our results are the following:

• (1)
  We prove that if M is a length metric space then ${\mathrm{Lip}}_0(M,X)$ has the Daugavet property. As a consequence, if M is length we obtain that $F\left(M\right){\hat{\otimes }}_{\pi }X$ has the Daugavet property. This gives an affirmative answer to [9, Question 1] (also asked in [16, 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 $F\left(M\right){\hat{\otimes }}_{\pi }X$ is octahedral, which solves [4, 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.