Invariants in convex geometry and Banach space theory

This workshop, sponsored by AIM and the NSF, will be devoted to invariants and their associated inequalities. Invariants and their associated inequalities play a central role in convex geometry and geometric analysis on the one side, and in Banach space theory on the other. Moreover, in recent years work is being done on various problems that are related to invariants from other fields. While there have been many interactions between the different communities in the last decade, no systematic effort has been devoted to specifically address interactions between the various invariants occurring in the two fields.

The aim of the workshop is to investigate invariants related to a few important problems at the intersection of geometric analysis and in Banach space theory.

Some of the topics that the workshop will focus on are the following:

• The ''local theory invariants'' of the projection bodies have been thoroughly investigated. However, this work has not been carried out for the the equally important class of intersection bodies. Among the problems to be address during the workshop are to find the right estimates for type, cotype, Gordon-Lewis constant, Banach-Mazur distances and entropy numbers for intersection and generalized intersection bodies.
• Another family of affine invariant quantities are the affine quermassintegrals and dual affine quermassintegrals introduced by E. Lutwak. Lutwak conjectured the extreme bodies that maximize/minimize these quantities although almost all of his questions are completely open. Moreover not even good asymptotic bounds are known for these quantities.
• The volume product is another invariant that the workshop wants to address. Blaschke and Santalo proved that the product of the volumes of a centrally symmetric, convex body and its polar is maximal for the Euclidean ball. Mahler conjectured that the minimum is attained for the cube and its polar, the crosspolytope.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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