The Interface between Convex Geometry and Harmonic Analysis
About this Title
Alexander Koldobsky, University of Missouri, Columbia, Columbia, MO and Vladyslav Yaskin, University of Oklahoma, Norman, OK
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2008: Volume 108
ISBNs: 978-0-8218-4456-4 (print); 978-1-4704-2468-8 (online)
MathSciNet review: MR2365157
MSC: Primary 52A20; Secondary 42A38
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.
Graduate students and research mathematicians interested in convex geometry, emphasizing methods from harmonic analysis.
Table of Contents
- Chapter 1. Hyperplane sections of $\ell _p$-balls
- Chapter 2. Volume and the Fourier transform
- Chapter 3. Intersection bodies
- Chapter 4. The Busemann-Petty 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