Softcover ISBN:  9780821844564 
Product Code:  CBMS/108 
List Price:  $36.00 
Individual Price:  $28.80 
eBook ISBN:  9781470424688 
Product Code:  CBMS/108.E 
List Price:  $34.00 
Individual Price:  $27.20 
Softcover ISBN:  9780821844564 
eBook: ISBN:  9781470424688 
Product Code:  CBMS/108.B 
List Price:  $70.00 $53.00 
Softcover ISBN:  9780821844564 
Product Code:  CBMS/108 
List Price:  $36.00 
Individual Price:  $28.80 
eBook ISBN:  9781470424688 
Product Code:  CBMS/108.E 
List Price:  $34.00 
Individual Price:  $27.20 
Softcover ISBN:  9780821844564 
eBook ISBN:  9781470424688 
Product Code:  CBMS/108.B 
List Price:  $70.00 $53.00 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 108; 2008; 107 ppMSC: Primary 52; 42; 44
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 BusemannPetty 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.
ReadershipGraduate students and research mathematicians interested in convex geometry, emphasizing methods from harmonic analysis.

Table of Contents

Chapters

Chapter 1. Hyperplane sections of $\ell _p$balls

Chapter 2. Volume and the Fourier transform

Chapter 3. Intersection bodies

Chapter 4. The BusemannPetty problem

Chapter 5. Projections and the Fourier transform

Chapter 6. Intersection bodies and $L_p$spaces

Chapter 7. On the road between polar projection bodies and intersection bodies

Chapter 8. Open problems


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

Chapters

Chapter 1. Hyperplane sections of $\ell _p$balls

Chapter 2. Volume and the Fourier transform

Chapter 3. Intersection bodies

Chapter 4. The BusemannPetty problem

Chapter 5. Projections and the Fourier transform

Chapter 6. Intersection bodies and $L_p$spaces

Chapter 7. On the road between polar projection bodies and intersection bodies

Chapter 8. Open problems