Delta-points in Banach spaces generated by adequate families

Abstract

We study delta-points in Banach spaces h A,p generated by adequate families A where 1 ≤ p < ∞. In the case the familiy A is regular and p = 1, these spaces are known as combinatorial Banach spaces. When p > 1 we prove that neither h A,p nor its dual contain delta-points. Under the extra assumption that 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 A consists of finite sets only we are able to rule out the existence of delta-points in h A,1 and Daugavet-points in its dual.We also show that if h A,1 is polyhedral, then it is either (I)polyhedral or (V)-polyhedral (in the sense of Fonf and Veselý).