**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: $32.00

Individual Price: $25.60

**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.

#### 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

#### Readership

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