Delta-points in Banach spaces generated by adequate families

We study delta-points in Banach spaces h(A,p) generated by adequate families A, where 1 <= p < infinity. 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 only of finite sets, 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 Vesely).