Triinu Veeorg (University of Tartu, Estonia) : A relative version of Daugavet points and Daugavet property
Résumé : We say that a norm one element $x$ in a Banach space $X$ is a emphrelative Daugavet-point if there exist $alpha>0$ and $x^*in S_X^*$ with $x^*(x)=1$ such that $sup_yin T|x-y|=2$ for every slice $T$ of $B_X$ which is contained in the slice $S(x^*,alpha)$. We show that these points lie strictly between Daugavet points and $Delta$-points. Furthermore, we provide a condition that a space with the Radon-Nikod’ym property must satisfy in order to be able to contain relative Daugavet points. We also study these points in absolute sums of Banach spaces and prove that the relative Daugavet property is different from both the Daugavet property and the diametral local diameter 2 property.
The talk is based on joint work with T. A. Abrahamsen, R. Aliaga, V. Lima, A. Martiny, Y. Perreau, and A. Prochazka.