Fourier analytic methods in convex geometry

August 20 to August 24, 2007

at the

American Institute of Mathematics, San Jose, California

organized by

Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch

Original Announcement

This workshop concerns the interface between convex geometry and harmonic analysis. Particular attention will be given to applications of Fourier analysis to the study of sections and projections of convex bodies.

The Fourier analytic approach to sections and projections of convex bodies has recently been developed and has led to several results including Fourier analytic solutions of the Busemann-Petty and Shephard problems, characterizations of intersection and projection bodies, extremal sections and projections of certain classes of bodies. The main idea of this approach is to express different properties of convex bodies in terms of the Fourier transform and then use methods of Fourier analysis to solve geometric problems.

One direction of the discussion will focus on the duality between sections and projections that remains one of the most intriguing mysteries of convex geometry. Many results on sections and projections are similar ("dual") to each other, but the proofs are usually very different. In many cases, the Fourier approach provides unified treatment of sections and projections, and it would be very interesting to explore the connections that are behind this phenomenon. Of particular interest is the question of whether the classes of intersection bodies and polar projection bodies are isomorphically equivalent.

Another direction is to try to extend the Fourier approach from isometric convex geometry to the asymptotic theory of convex bodies, where one is mostly interested in phenomena occurring when the dimension goes to infinity. This theory has numerous applications to functional analysis, probability, computer science.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Slides from workshop talks

Geometric Tomography by Gardner

Generalized intersection bodies and... by Milman

Sums of similar convex bodies and spherical harmonics by Schneider

Directed sections and projection functions by Weil (10 Meg scanned slides)

Papers arising from the workshop:

Uniform estimates for order statistics and Orlicz functions
by  Yehoram Gordon, Alexander Litvak, Carsten Schutt, and Elisabeth Werner