Delta-points in Banach spaces generated by adequate families

Abstract

We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by adequate families $\mathcal{A}$ where $1\le p<\infty$. In the case the familiy $\mathcal{A}$ is regular and $p=1,$ these spaces are known as combinatorial Banach spaces. When $p>1$ we prove that neither $h_{\mathcal{A},p}$ nor its dual contain delta-points. Under the extra assumption that $\mathcal{A}$ is regular, we prove that the same is true when $p=1.$ In particular the Schreier spaces and their duals fail to have delta-points. If $\mathcal{A}$ consists of finite sets only we are able to rule out the existence of delta-points in $h_{\mathcal{A},1}$ and Daugavet-points in its dual. We also show that if $h_{\mathcal{A},1}$ is polyhedral, then it is either (I)-polyhedral or (V)-polyhedral (in the sense of Fonf and Veselý).

Fixed some misprints. Added Examples 2.4, 2.5 and Corollary 5.4