**Mathematical Surveys and Monographs**

Volume: 116;
2005;
170 pp;
Softcover

MSC: Primary 52; 46;
Secondary 42; 60

**Print ISBN: 978-1-4704-1952-3
Product Code: SURV/116.S**

List Price: $74.00

AMS Member Price: $59.20

MAA Member Price: $66.60

**Electronic ISBN: 978-1-4704-1343-9
Product Code: SURV/116.E**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

#### Supplemental Materials

# Fourier Analysis in Convex Geometry

Share this page
*Alexander Koldobsky*

The study of the geometry of convex bodies based on information about sections
and projections of these bodies has important applications in many areas of
mathematics and science. In this book, a new Fourier analysis approach is
discussed. The idea is to express certain geometric properties of bodies in
terms of Fourier analysis and to use harmonic analysis methods to solve
geometric problems.

One of the results discussed in the book is Ball's theorem, establishing the
exact upper bound for the \((n-1)\)-dimensional volume of hyperplane sections of
the \(n\)-dimensional unit cube (it is \(\sqrt{2}\) for each \(n\geq 2\)). Another is
the Busemann–Petty problem: if \(K\) and \(L\) are two convex origin-symmetric
\(n\)-dimensional bodies and the \((n-1)\)-dimensional volume of each central
hyperplane section of \(K\) is less than the \((n-1)\)-dimensional volume of the
corresponding section of \(L\), is it true that the \(n\)-dimensional volume of \(K\)
is less than the volume of \(L\)? (The answer is positive for \(n\le 4\) and
negative for \(n>4\).)

The book is suitable for graduate students and researchers interested in
geometry, harmonic and functional analysis, and probability. Prerequisites for
reading this book include basic real, complex, and functional analysis.

#### Readership

Graduate students and research mathematicians interested in Fourier analysis and geometry.

#### Table of Contents

# Table of Contents

## Fourier Analysis in Convex Geometry

- Contents v6
- Chapter 1. Introduction 18
- Chapter 2. Basic Concepts 1320
- Chapter 3. Volume and the Fourier Transform 4956
- Chapter 4. Intersection Bodies 7178
- Chapter 5. The Busemann-Petty Problem 95102
- Chapter 6. Intersection Bodies and L[sub(p)]-Spaces 115122
- Chapter 7. Extremal Sections of l[sub(q)]-Balls 143150
- Chapter 8. Projections and the Fourier Transform 151158
- Bibliography 163170
- Index 169176