This work was supported by MCIN/AEI/10.13039/501100011033: Grant PID2021-122126NB-C31 and by Junta de Andalucía: Grants FQM-0185 and PY20_00255. The research of Rubén Medina was also supported by FPU19/04085 MIU (Spain) Grant, by GA23-04776S project (Czech Republic) and by SGS22/053/OHK3/1T/13 project (Czech Republic). The research of Abraham Rueda Zoca was also supported by Fundación Séneca: ACyT Región de Murcia grant 21955/PI/22 and by Generalitat Valenciana project CIGE/2022/97.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.MCIN/AEI/10.13039/501100011033 PID2021-122126NB-C31Junta de Andalucía FQM-0185, PY20_00255FPU19/04085 MIU (Spain)Czech Republic GA23-04776S, SGS22/053/OHK3/1T/13Fundación Séneca: ACyT Región de Murcia grant 21955/PI/22Generalitat Valenciana CIGE/2022/9
