Separated sets and Auerbach systems in Banach spaces

Abstract

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space X X , the unit sphere S X S_X always contains an uncountable ( 1 + ) (1+) -separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c 0 ( ω 1 ) c_0(\omega _1) . Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically ( 1 + ε ) (1+\varepsilon ) -separated subset of any regular cardinality not exceeding the density of X X ; should the space X X be super-reflexive, the unit sphere of X X contains such a subset of cardinality equal to the density of X X . The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.