Two refinements of the Bishop--Phelps--Bollobás modulus

Abstract

Extending the celebrated result by Bishop and Phelps that the set of norm attaining functionals is always dense in the topological dual of a Banach space, Bollobás proved the nowadays known as the Bishop--Phelps--Bollobás theorem, which allows to approximate at the same time a functional and a vector in which it almost attains the norm. Very recently, two Bishop--Phelps--Bollobás moduli of a Banach space have been introduced $[\textit{J. Math. Anal. Appl.}~412 (2014), 697--719]$ to measure, for a given Banach space, what is the best possible Bishop--Phelps--Bollobás theorem in this space. In this paper we present two refinements of the results of that paper. On the one hand, we get a sharp general estimation of the Bishop--Phelps--Bollobás modulus as a function of the norms of the point and the functional, and we also calculate it in some examples, including Hilbert spaces. On the other hand, we relate the modulus of uniform non-squareness with the Bishop--Phelps--Bollobás modulus obtaining, in particular, a simpler and quantitative proof of the fact that a uniformly non-square Banach space cannot have the maximum value of the Bishop--Phelps--Bollobás modulus.