Polyhedrality and decomposition

TL;DR

This paper presents two results that simplify the process of finding equivalent polyhedral norms on Banach spaces with specific basis properties, using sphere decompositions into countable pieces.

Contribution

It introduces new criteria based on sphere decompositions for establishing the existence of equivalent polyhedral norms on certain Banach spaces.

Findings

Provides conditions under which Banach spaces admit polyhedral norms

Includes examples of spaces with equivalent polyhedral norms

Simplifies the process of identifying polyhedral norms

Abstract

The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The hypotheses of both results are based on decomposing the unit sphere of a Banach space into countably many pieces, such that each one satisfies certain properties. Some examples of spaces having equivalent polyhedral norms are given.