Almost flat locally finite coverings of the sphere
Loosely speaking, a subset of the unit sphere of a Banach space is called flat if its corresponding set of support functionals has small diameter. This concept is naturally connected with subdifferentials of the norm. The main result is that a Banach space has separable dual if and only if it can be renormed so that its new unit sphere admits a countable locally finite tiling by uniformly flat sets. For bigger Banach spaces, the existence of a similar covering is equivalent to the gap between Gâteaux and Fréchet smoothness of the norm.
scientific article; zbMATH DE number 2145885
scientific article