The Interface between Convex Geometry and Harmonic Analysis (CBMS Regional Conference Series in Mathematics, 108) - Softcover

Alexander Koldobsky And Vladyslav Yaskin

Softcover

ISBN 10: 0821844563 ISBN 13: 9780821844564

Publisher: American Mathematical Society, 2007

Synopsis

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.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2007
• Language: English
• ISBN 10: 0821844563
• ISBN 13: 9780821844564
• Binding: Paperback
• Number of pages: 107