Smallness and the covering of a Banach space (Q1931676)

A subset \(A\) of a metric space is small (in \textit{J. Arias de Reyna's} terminology from [Proc. Edinb. Math. Soc., II. Ser. 31, No. 2, 217--229 (1988; Zbl 0662.46021)]) if there is a sequence of balls \(B(x_n, r_n)\) with \(r_n \to 0\) such that for every \(m \in \mathbb N\) one has \(A \subset \bigcup_{n \geq m}B(x_n, r_n)\). The authors give a number of generalizations and simplifications of several previously known results on small sets in Banach spaces. In particular, they prove that a Banach space \(X\) can be equivalently renormed so that the set of extreme points of the dual unit ball \(B_{X^*}\) is small if and only if it can be renormed so that the set of extreme points of \(B_{X^*}\) is countable, thus improving significantly a theorem by \textit{E. Behrends} and \textit{V. M. Kadets} [Stud. Math. 148, No. 3, 275--287 (2001; Zbl 1002.46012)]. Applications to isomorphically polyhedral spaces are given.