# Mahler's conjecture and duality in convex geometry

August 9 to August 13, 2010

at the

American Institute of Mathematics, San Jose, California

organized by

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

## Original Announcement

This workshop 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 $\R^n$ is defined by $P(K)=vol_n(K)vol_n(K^*)$ 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.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Isomorphic properties of Intersection bodies
by  A. Koldobsky, G. Paouris, and M. Zymonopoulou
A problem of Klee on inner section functions of convex bodies
by  Richard J. Gardner, Dmitri Ryabogin, Vladyslav Yaskin, and Artem Zvavitch,  J. Differential Geom. 91 (2012), no. 2, 261-279  MR2971289