Delta-points in Banach spaces generated by adequate families

TL;DR

This paper investigates the presence of delta-points in Banach spaces generated by adequate families, showing their absence in many cases and characterizing polyhedral structures.

Contribution

It establishes conditions under which delta-points do not exist in these Banach spaces and characterizes polyhedrality types.

Findings

Delta-points are absent in $h_{ ext{A},p}$ for $p>1$ and regular $ ext{A}$ when $p=1$.

Schreier spaces and their duals lack delta-points.

Polyhedrality of $h_{ ext{A},1}$ implies it is either (I)- or (V)-polyhedral.

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 Vesel\'y).