The Daugavet property in spaces of vector-valued Lipschitz functions

Abstract

We prove that if a metric space M has the finite CEP then F(M)⊗ˆπX has the Daugavet property for every non-zero Banach space X. This applies, for instance, if M is a Banach space whose dual is isometrically an L1(μ) space. If M has the CEP then L(F(M),X)=Lip0(M,X) has the Daugavet property for every non-zero Banach space X, showing that this is the case when M is an injective Banach space or a convex subset of a Hilbert space.