**CBMS Regional Conference Series in Mathematics**

Volume: 108;
2008;
107 pp;
Softcover

MSC: Primary 52; 42; 44;

**Print ISBN: 978-0-8218-4456-4
Product Code: CBMS/108**

List Price: $34.00

Individual Price: $27.20

**Electronic ISBN: 978-1-4704-2468-8
Product Code: CBMS/108.E**

List Price: $32.00

Individual Price: $25.60

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#### Supplemental Materials

# The Interface between Convex Geometry and Harmonic Analysis

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*Alexander Koldobsky; Vladyslav Yaskin*

A co-publication of the AMS and CBMS

The study of convex bodies is a central part of geometry, and is
particularly useful in applications to other areas of mathematics and the
sciences. Recently, methods from Fourier analysis have been developed that
greatly improve our understanding of the geometry of sections and
projections of convex bodies. The idea of this approach is to express
certain properties of bodies in terms of the Fourier transform and then to
use methods of Fourier analysis to solve geometric problems. The results
covered in the book include an analytic solution to the Busemann-Petty
problem, which asks whether bodies with smaller areas of central
hyperplane sections necessarily have smaller volume, characterizations of
intersection bodies, extremal sections of certain classes of bodies, and a
Fourier analytic solution to Shephard's problem on projections of convex
bodies.

The book is written in the form of lectures accessible to graduate
students. This approach allows the reader to clearly see the main ideas
behind the method, rather than to dwell on technical difficulties. The
book also contains discussions of the most recent advances in the subject.
The first section of each lecture is a snapshot of that lecture. By
reading each of these sections first, novices can gain an overview of the
subject, then return to the full text for more details.

#### Readership

Graduate students and research mathematicians interested in convex geometry, emphasizing methods from harmonic analysis.

#### Table of Contents

# Table of Contents

## The Interface between Convex Geometry and Harmonic Analysis

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Hyperplane sections of l[sub(p)]-balls 112 free
- Chapter 2. Volume and the Fourier transform 920
- Chapter 3. Intersection bodies 2132
- Chapter 4. The Busemann-Petty problem 3950
- Chapter 5. Projections and the Fourier transform 5970
- Chapter 6. Intersection bodies and L[sub(p)]-spaces 6778
- Chapter 7. On the road between polar projection bodies and intersection bodies 7586
- Chapter 8. Open problems 8798
- Bibliography 101112
- Index 107118
- Back Cover Back Cover1122