Delta- and Daugavet-points in Banach spaces

Trond Arnold Abrahamsen; Rainis Haller; Vegard Lima; Katriin Pirk

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arXiv · 1812.02450

Delta- and Daugavet-points in Banach spaces

Trond Arnold Abrahamsen·Rainis Haller·Vegard Lima·Katriin Pirk

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Abstract

A $\Delta$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if and only if every norm one element is a Daugavet-point. We show that $\Delta$- and Daugavet-points are the same in $L_1$-spaces, $L_1$-preduals, as well as in a big class of M\"untz spaces. We also provide an example of a Banach space where all points on the unit sphere are $\Delta$-points, but where none of them are Daugavet-points. We also study the property that the unit ball is the closed convex hull of its $\Delta$-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all $C(K)$ spaces, $K$ infinite compact Hausdorff, as well as all M\"untz spaces have this property. Moreover, we show that this property is stable under absolute sums.

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BibTeX RIS

Trond Arnold Abrahamsen, Rainis Haller, Vegard Lima, Katriin Pirk. 2018-12-06. Delta- and Daugavet-points in Banach spaces. https://arxiv.org/abs/1812.02450

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