This is an announcement for the paper "Polyhedrality in Pieces" by V. P. Fonf, A. J. Pallares, R. J. Smith, and S. Troyanski.

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 fi?nal section.

Archive classification: math.FA

Submitted from: apall@um.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1302.0160

or

http://arXiv.org/abs/1302.0160