We study Daugavet- and $$\Delta$$ Δ -points in Banach spaces. A norm one element x is a Daugavet-point (respectively, a $$\Delta$$ Δ -point) if in every slice of the unit ball (respectively, in every slice of the unit ball containing x) you can find another element of distance as close to 2 from x as desired. In this paper, we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $$\Delta$$ Δ -point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $$\Delta$$ Δ -points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally, we prove that there exists a superreflexive Banach space with a Daugavet- or $$\Delta$$ Δ -point provided there exists such a space satisfying a weaker condition.

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal ω. We also gather results linking these notions with the spreading models of the space. IntroductionLet (T N ) N ≥1 be the family of countably branching trees endowed with the hyperbolic distance. The main result in the present paper is the following theorem.Let us briefly recall the context and the motivation of this theorem. In 1986, J. Bourgain gave in his paper [6] a metric invariant characterizing super-reflexivity: the non equi-Lipschitz embeddability of the family (D N ) N ≥1 of dyadic trees endowed with the hyperbolic distance. His result is the following. Theorem 1.2. Let X be a Banach space. Then X is super-reflexive if and only if the family (D N ) N ≥1 does not equi-Lipschitz embed into X.This was the first step in the so called Ribe program which looks for metric invariants characterizing local properties of Banach spaces. The reader can have a look at [18] for a detailed introduction to the Ribe program and for a survey of results in this direction. A short proof of the non-embeddability of the family of dyadic trees into a super-reflexive space was given more recently by R. Kloeckner in [11] using uniform convexity and a self-improvement argument.In [2], F. Baudier, N. J. Kalton and G. Lancien introduced a new metric invariant in order to give a metric characterization of asymptotic properties of Banach spaces: the non equi-Lipschitz embeddability of the family (T N ) N ≥1 of countably branching trees endowed with the hyperbolic distance. The main tool in their paper is a derivation index called Slzenk index. We will introduce these objects in section 2. They proved the following results.2010 Mathematics Subject Classification. 46B20, 46B80, 46B85, 46T99. The author is supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03). 1 2 Y. PERREAU Theorem 1.3. Let X be a separable Banach space. If S Z (X) > ω or if S Z (X * ) > ω, then the family (T N ) N ≥1 equi-Lipschitz embeds into X. Theorem 1.4. Let X be a reflexive separable Banach space. If S Z (X) ≤ ω and S Z (X * ) ≤ ω, then the family (T N ) N ≥1 does not equi-Lipschitz embed into X.Note that the assumption of separability can be removed in both theorems by using properties of the Szlenk index. Using an argument à la Kloeckner and the property (β) of Rolewicz, F. Baudier and S. Zhang gave in [4] a shorter proof of the second theorem. However, this argument cannot be extended to a more general setting since (β) implies reflexivity.In [10], N. J. Kalton applied results coming from the study of Orlicz sequence spaces to get estimates on the spreading models of Banach spaces which coarse-Lipschitz embed into asymptotically uniformly convex spaces. Inspired by this method, we will give in section 4 estima...

We study Daugavet-and ∆-points in Banach spaces. A norm one element x is a Daugavet-point (respectively a ∆-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing x) you can find another element of distance as close to 2 from x as desired.In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet-or ∆-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain ∆-points. We also show that the same conclusion holds true for the James tree space as well as for its predual.Finally we prove that there exists a superreflexive Banach space with a Daugavet-or ∆-point provided there exists such a space satisfying a weaker condition.