We study the Daugavet property in the space of Lipschitz functions Lip 0 (M ) for a complete metric space M . Namely we show that Lip 0 (M ) has the Daugavet property if and only if M is a length space. This condition also characterises the Daugavet property in the Lipschitz free space F(M ). Moreover, when M is compact, we show that either F(M ) has the Daugavet property or its unit ball has a strongly exposed point. If M is an infinite compact subset of a strictly convex Banach space then the Daugavet property of Lip 0 (M ) is equivalent to the convexity of M .2010 Mathematics Subject Classification. Primary 46B20; Secondary 54E50. Key words and phrases. Daugavet property; space of Lipschitz functions: Lipschitz-free space; length space; strongly exposed point. 2 L. GARCÍA-LIROLA, A. PROCHÁZKA AND A. RUEDA ZOCAthat Lip 0 (M ) has the Daugavet property whenever M is a length metric space.Here we prove the converse implication, thus obtaining our main theorem (Theorem 3.3) which completely characterises those complete metric spaces M such that Lip 0 (M ) has the Daugavet property. As a consequence of Theorem 3.3 we also get that the space Lip 0 (M ) has the Daugavet property if, and only if, its canonical predual F(M ) (see the formal definition below) has the Daugavet property, extending the corresponding result in the compact case which was proved in [19].This paper is organised as follows. In Section 2 we introduce necessary definitions and establish several results concerning length and geodesic metric spaces, in particular we show that a complete local space is a length space. We also study sufficient conditions for a metric space to be geodesic. Section 3 is devoted to the proof of the main theorem, the charaterisation of Lipschitz free spaces and spaces of Lipschitz functions with the Daugavet property. Section 4 includes a characterisation of strongly exposed points in B F (M ) (Theorem 4.4). We use this result to prove in Corollary 4.11 that, when M is compact, the Daugavet property of F(M ) is equivalent to the absence of strongly exposed points of B F (M ) . It is not clear whether the absence of strongly exposed points of B F (M ) implies in general that M is a length space. In the first part of Section 5 we gather some partial evidence to support such a conjecture. In the second part of Section 5 we study the Daugavet property in the spaces of vector-valued functions Lip 0 (M, X). This is used to give new examples of spaces of linear bounded operators and of projective tensor products enjoying the Daugavet property.Notation: Throughout the paper we will only consider real Banach spaces. Given a Banach space X, we will denote the closed unit ball and the unit sphere of X by B X and S X respectively. We will also denote by X * the topological dual of X.By a slice of the unit ball B X of a Banach space X we will mean a set of the following formwhere f ∈ S X * and α > 0. Notice that slices are non-empty relatively weakly open and convex subsets of B X whose complement is also convex.Given a metric space ...

We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say x and y, which define it satisfy that the metric segment [x, y] only contains x and y. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about normattainment in spaces of vector valued Lipschitz functions.

We show that the class of Lipschitz-free spaces over closed subsets of any complete metric space M is closed under arbitrary intersections, improving upon the previously known finite-diameter case. This allows us to formulate a general and natural definition of supports for elements in a Lipschitz-free space F (M ). We then use this concept to study the extremal structure of F (M ). We prove in particular that (δ(x) − δ(y))/d(x, y) is an exposed point of the unit ball of F (M ) whenever the metric segment [x, y] is trivial, and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself. We also characterise the extreme points of the positive unit ball: they are precisely the normalized evaluation functionals on points of M .2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B04, 54E50.