Daugavet- and delta-points in Banach spaces with unconditional bases

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 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 $2$. 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 Daugavet-points, and a Banach space with a $1$-unconditional basis with a unit ball in which the Daugavet-points are weakly dense.

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