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.

We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1 1 -unconditional basis. A norm one element x x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 2 from x x . A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2 2 . We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1 1 -unconditional basis with delta-points, but with no Daugavet-points, and a Banach space with a 1 1 -unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

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We study the existence of Daugavet-and delta-points in the unit sphere of Banach spaces with a 1-unconditional basis. A norm one element x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from x. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is wellknown that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an unconditional basis with suppression unconditional constant strictly less than 2.We show that no Banach space with a subsymmetric basis can have delta-points. In contrast we construct a Banach space with a 1-unconditional basis with delta-points, but with no Daugavetpoints, and a Banach space with a 1-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

Abstract:We show that Müntz spaces, as subspaces of C [ , ], contain asymptotically isometric copies of c and that their dual spaces are octahedral.