# July 29 - August 3, 2006

 This conference is on the interface between convex geometry and harmonic analysis. Alexander Koldobsky will deliver ten lectures on a number of topics of common interest to both harmonic analysts and geometers. This web site now includes links to lectures of the videos and a picture of the participants. See below.

Lecturer: Professor Alexander Koldobsky, University of Missouri-Columbia

Organizers:
David Auckly
Dmitry Ryabogin

# Participants

Row 1 - East: Paouris Grigoris, Marianne Korten, Julie Bergner, Harrison Potter, Jennifer Anderson, Dov Rhodes, Maryna Yaskina, Vlad Yaskin, Denys Maslov, Timothy Kohler, Eyuel Abebe, Jameson Graber

Row 2 - East: Dave Auckly, Alex Iosevich, Chuck Moore, Dennis Hall, Ed White, Diego Maldonado, Wes Cross, Wolfgang Weil, Dmitry Ryabogin, Yuriy Kolomiets, Bob Burkel

Row 3 - East: Sapto Indratno, Pietro Poggi-Corradini, Alex Koldobsky, Yehoram Gordon, Fedja Nazaron, Chris Shane, Virginia Naibo, Maria Angeles Alfonseca, Donald Adongo, Artem Zvavitch, Jeff Schlaerth, Paul Goodey

# The lectures titles/abstracts (including links to the videos/notes) are:

1. Hyperplane sections and projections of $l_p$-balls.

We derive the Fourier transform formulas for the volume of hyperplane sections and projections of $l_p$-balls and apply them to find the extremal sections and projections of these balls. We review the results of Hensley, Vaaler, Ball, Meyer and Pajor, Oleszkiewicz and Pelczynski, Barthe and Naor.

We did not get a video of this lecture, but you can read notes at: Lecture 1 notes

2. Volume and the Fourier transform: general formulas.

We prove several formulas relating the volume of sections of convex bodies to the Fourier transform of powers of the corresponding norm. These formulas are crucial for the Fourier analytic approach to sections of convex bodies.

3. Intersection bodies.

We study the class of intersection bodies, introduced by Lutwak and playing an important role in the dual Brunn-Minkowski theory. We establish a Fourier analytic characterization of intersection bodies and apply it to certain classes of bodies.

4. The Busemann-Petty problem: solution and generalizations.

The Busemann-Petty problem asks whether origin symmetric convex bodies with uniformly smaller areas of central hyperplane sections necessarily have smaller volume. We review the solution and consider several generalizations.

5. Projections of convex bodies and the Fourier transform.

We show that the Fourier approach to sections of convex bodies can be extended to projections. In particular, we consider Shephard's problem, which is the projection analog of the Busemann-Petty problem, and show that both problems can be given very similar solutions.

6. Positive definite functions and stable random vectors.

We show the connection between positive definite norm dependent functions and stable random vectors, with emphasis on Schoenberg's problems and their generalizations.

7. Isometric embedding of normed spaces in $L_p$, $-\infty < p <\infty.$

We study the connections between positive definite distributions and embedding of normed spaces in $L_p,$ including the case $p\le 0,$ where the concept of embedding is defined as analytic continuation of the corresponding property for $p>0.$

8. A functional analytic interpretation of intersection and polar projection bodies.

We show that polar projection bodies are the unit balls of subspaces of $L_1,$ and $k$-intersection bodies are the "unit balls of subspaces" of $L_{-k}.$ We study the intermediate classes of bodies, providing a continuous path from polar projection bodies to interection bodies.

9. Duality of sections and projections.

It has been known for a long time that many results on sections and projections are dual to each other, while the proofs are completely different. We study this phenomenon from the point of view of the conjecture that intersection and polar projection bodies are isomorphically equivalent.

10. New directions and open problems.