#
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: