Interplay of convex geometry and Banach space theory

There are traditionally many interactions between the convex geometry community and the Banach space community. In recent years, work is being done as well on problems that are related to notions and concepts from other fields. The interaction of convex geometry and Banach space theory, and also with other areas, is due to high dimensional phenomena which lie at the crossroad of convex geometry and Banach space theory. But even though there are close ties between the two communities, we felt that much more could be achieved for both areas by having a fresh look, together, at some of the main problems of the two areas. That was exactly the topic of the workshop. Banach space theory consists essentially of two subfields, the local theory of Banach spaces and the infinite dimensional theory of Banach spaces. Of course, there are interactions between the two. Relevant for the conference was only the local theory of Banach spaces. There, one tries to find out to what extend the finite dimensional subspaces of a given Banach space determine the nature of the Banach space. Of particular importance to decide on the last mentioned aspect is the role played by invariances. We give a simple example that explains this. The type p constant of a Banach space X [30] is the infimum over all constants C such that for all n and all x1, . . . , xn ∈ X ∫ 1 0 ∥∥∥∥ n ∑