Fourier Analysis in Convex Geometry
Seiten
2005
|
illustrated Edition
American Mathematical Society (Verlag)
978-0-8218-3787-0 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-3787-0 (ISBN)
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Discusses Fourier analysis approach. This book expresses certain geometric properties of bodies in terms of Fourier analysis and uses harmonic analysis methods to solve geometric problems. It is suitable for graduate students and researchers interested in geometry, harmonic and functional analysis, and probability.
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 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.
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 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.
Introduction Basic concepts Volume and the Fourier transform Intersection bodies The Busemann-Petty problem Intersection bodies and $L_p$-spaces Extremal sections of $/ell_q$-balls Projections and the Fourier transform Bibliography Index.
| Erscheint lt. Verlag | 1.6.2005 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Zusatzinfo | Illustrations |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 0-8218-3787-7 / 0821837877 |
| ISBN-13 | 978-0-8218-3787-0 / 9780821837870 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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