Mahler's conjecture and duality in convex geometry

August 9 to August 13, 2010

at the

American Institute of Mathematics, Palo Alto, California

organized by

Alexander Koldobsky, Dmitry Ryabogin, Vladyslav Yaskin, and Artem Zvavitch

This workshop, sponsored by AIM and the NSF, will be devoted to duality problems in convex geometry, which deal with relations between convex bodies and their polar bodies. The participants will explore the opportunities opened by a flurry of recent results related to the problem, most of which are based on new promising analytic techniques.

The main topics for the workshop are:

  1. The volume product of a convex body K in Rn is defined by P(K)= Voln(K) Voln(K*), where K* is the polar body of K. Mahler's conjecture asks whether the minimum of the volume product in the class of origin-symmetric convex bodies is attained at the unit cube. Despite many important partial results, the problem is still open in dimensions 3 and higher. The participants will explore the opportunities opened by new proofs of the Bourgain-Milman theorem (establishing Mahler's conjecture up to an absolute constant).
  2. It has been known for a long time that many results on sections and projections of convex bodies are dual to each other, in the sense that sections of a body behave in a similar way to projections of the polar body. Methods of Fourier analysis can be applied to develop a unified approach to some of these results. The participants will try to extend this Fourier approach to other problems, in particular to the question of whether intersection and polar projection bodies are isomorphically equivalent.

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.

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