Abstract. We introduce two Bishop-Phelps-Bollobás moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobás theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and an inequality with respect to duality. We calculate the two moduli for Hilbert spaces and also present many examples for which the moduli have the maximum possible value (among them, there are C(K) spaces and L 1 (µ) spaces). Finally, we show that if a Banach space has the maximum possible value of any of the moduli, then it contains almost isometric copies of the real space ∞ and present an example showing that this condition is not sufficient.

We study the rank-one numerical index of a Banach space, namely the inmum of the numerical radii of those rank-one operators on the space which have norm-one. We show that the rank-one numerical index is always greater or equal than 1/ e. We also present properties of this index and some examples.

Geostatistics offers various techniques of estimation and simulation that have been satisfactorily applied in solving geological problems. In this sense, conditional geostatistical simulation is applied to calculate the probability of occurrence of an earthquake with a lower than or equal magnitude to one determined during a seismic series. It is possible to calculate the energy of the next most probable earthquake from a specific time, given knowledge of the structure existing among earthquakes occurring prior to a specific moment.

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 [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.

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