Fourier Analysis in Convex Geometry (Mathematical Surveys and Monographs) - Softcover

Koldobsky, Alexander

Softcover

ISBN 10: 1470419521 ISBN 13: 9781470419523

Publisher: American Mathematical Society, 2014

Synopsis

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 all mathematicians interested in geometry, harmonic and functional analysis, and probability. Prerequisites for reading this book include basic real, complex, and functional analysis.

About this title

About the Author

Alexander Koldobsky, University of Missouri, Columbia, MO, USA.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2014
• Language: English
• ISBN 10: 1470419521
• ISBN 13: 9781470419523
• Binding: Paperback
• Number of pages: 170