Softcover ISBN:  9781470419523 
Product Code:  SURV/116.S 
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AMS Member Price:  $103.20 
eBook ISBN:  9781470413439 
Product Code:  SURV/116.E 
List Price:  $125.00 
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AMS Member Price:  $100.00 
Softcover ISBN:  9781470419523 
eBook: ISBN:  9781470413439 
Product Code:  SURV/116.S.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 
Softcover ISBN:  9781470419523 
Product Code:  SURV/116.S 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413439 
Product Code:  SURV/116.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470419523 
eBook ISBN:  9781470413439 
Product Code:  SURV/116.S.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 116; 2005; 170 ppMSC: Primary 52; 46; Secondary 42; 60
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 \((n1)\)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 originsymmetric \(n\)dimensional bodies and the \((n1)\)dimensional volume of each central hyperplane section of \(K\) is less than the \((n1)\)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.
ReadershipGraduate students and research mathematicians interested in Fourier analysis and geometry.

Table of Contents

Chapters

1. Introduction

2. Basic concepts

3. Volume and the Fourier transform

4. Intersection bodies

5. The BusemannPetty problem

6. Intersection bodies and $L_p$spaces

7. Extremal sections of $\mathscr {\ell }_q$balls

8. Projections and the Fourier transform


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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 \((n1)\)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 originsymmetric \(n\)dimensional bodies and the \((n1)\)dimensional volume of each central hyperplane section of \(K\) is less than the \((n1)\)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.
Graduate students and research mathematicians interested in Fourier analysis and geometry.

Chapters

1. Introduction

2. Basic concepts

3. Volume and the Fourier transform

4. Intersection bodies

5. The BusemannPetty problem

6. Intersection bodies and $L_p$spaces

7. Extremal sections of $\mathscr {\ell }_q$balls

8. Projections and the Fourier transform