Polyhedrality in Pieces

TL;DR

This paper introduces two theorems that simplify the process of identifying equivalent polyhedral norms on Banach spaces, utilizing countable decompositions of spheres or dual subsets, and provides new examples of such spaces.

Contribution

It presents two new tools, Theorems 4 and 7, that facilitate the detection of polyhedral norms and unifies existing results for specific Banach spaces.

Findings

Theorems 4 and 7 provide necessary and sufficient conditions for polyhedral renorming.

Unified results for C(K) spaces and spaces with unconditional bases.

New examples of spaces with equivalent polyhedral norms.

Abstract

The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere S_X or of certain subsets of the dual ball of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K) spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the final section.