Polytope Approximations of the Unit Ball of ℓnp

Summary

A simple and explicit method is given for approximating the unit ball of ℓⁿ_p by polytopes. The method leads to a natural generalization of £p-spaces with good duality and interpolation properties.

1. Introduction

The classical spaces ℓ_p and L_p are the best known and in many ways most fundamental examples of Banach spaces. In view of their interesting properties it is natural to ask whether the role of the function t^P in these spaces can be played by other more general functions. This question was answered by Orlicz, who defined a certain class of functions, now known as Orlicz functions, and associated with each one a sequence space and a function space, now called an Orlicz sequence space and Orlicz function space. The Orlicz spaces are generally regarded as the correct and most natural spaces to associate with given Orlicz functions.

One of the aims of this paper is to cast doubt on that view, at least in its isometric interpretation. We shall do this by discussing a different generalization which arises geometrically and has two desirable isometric properties lacked by Orlicz spaces. First, the dual of one of our spaces is isometric to another such space. Second, complex interpolation between two of our spaces yields a third in a natural way. Irritatingly, we have not managed to establish whether our new spaces are isomorphic to Orlicz spaces, in which case they are a useful renorming of them, or whether they are completely different. Our route to the new generalization starts with an unusual (perhaps even eccentric) problem which will be described below, and which relates more to the polytope approximations of the title.