ENGLISH

Fourier Analysis In Convex Geometry

By Alexander Koldobsky

Book information

Publisher
American Mathematical Society,
Year
2005
ISBN
0821837877, 9780821837870
Open Library ID
OL9595375M
Language
english
Format
PDF
Filesize
13 MB (13965532 bytes)
Series
Mathematical Surveys and Monographs
Pages
180\180
Library
avaxhome.ws
Scanned
yes
Time added
2010-05-31 15:29:46

Description

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.

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