9 - Various numerical parameters

Summary

As is abundantly clear from the previous chapters there is a huge variety of non-isometric Minkowski spaces in each dimension. It is not feasible to seek numerical invariants that will completely classify all these spaces. However, there are a variety of properties that stand out as being significant in the theory of infinite dimensional Banach spaces. It frequently happens that these properties can be framed in terms of numerical parameters that depend on the “shape” of the space and are invariant under isometries. The projection constant in §9.1 below is a good example of this situation. In finite dimensional spaces such numerical parameters lead to several interesting questions. Firstly, one can try to calculate the exact value of the parameter for particular spaces. This is often a difficult problem. Secondly, one can ask for the bounds, as precisely as possible, of the parameter over all Minkowski spaces of a fixed dimension. Instances of this problem were considered in §6.5 and §7.4, where bounds for the parameter μ_B(∂B) were given for each of the two definitions of μ_B. Thirdly, one can investigate the asymptotic behaviour of the parameter as the dimension gets large. The asymptotic behaviour is often significant for the infinite dimensional theory. Finally, one can use differences in the value of a particular parameter for two distinct spaces to measure how far apart (in the Banach–Mazur or some other metric) the two spaces are. Various examples of these four problems will be considered in this last chapter.