This is an announcement for the paper “Delta- and Daugavet-points in Banach spaces” by Trond Arnold Abrahamsenhttps://arxiv.org/search/math?searchtype=author&query=Abrahamsen%2C+T+A, Rainis Hallerhttps://arxiv.org/search/math?searchtype=author&query=Haller%2C+R, Vegard Limahttps://arxiv.org/search/math?searchtype=author&query=Lima%2C+V, Katriin Pirkhttps://arxiv.org/search/math?searchtype=author&query=Pirk%2C+K. 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 Müntz 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 Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.02450