Fourier Analysis in Convex GeometryThe 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. |
Contents
| 1 | |
Chapter 2 Basic Concepts | 13 |
Chapter 3 Volume and the Fourier Transform | 49 |
Chapter 4 Intersection Bodies | 71 |
Chapter 5 The BusemannPetty Problem | 95 |
Chapter 6 Intersection Bodies and LsubpSpaces | 115 |
Chapter 7 Extremal Sections of lsubqBalls | 143 |
Chapter 8 Projections and the Fourier Transform | 151 |
| 163 | |
| 169 | |
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Common terms and phrases
affirmative argument Busemann-Petty problem central hyperplane section constant continuous function converges cube defined dimension eaſists embedding in Lp embeds in Lp embeds isometrically equal Euclidean ball exists a finite fact finite Borel measure Fourier transform Fubini theorem function f function of degree Hölder's inequality homogeneous function homogeneous of degree hyperplane sections inequality infinitely smooth origin-symmetric integrable function intersection bodies k-intersection Lemma locally integrable Lp-spaces Lutwak measure pu Minkowski functional n-dimensional negative non-negative normed space origin-symmetric convex body origin-symmetric star body parallel section function Parseval's formula positive definite distribution positive definite functions projection body prove radial metric Radon transform random variables result follows right-hand side second derivative Shephard's problem ſº solution sphere S"T spherical harmonics spherical Radon transform subspace of Lp Suppose test function Theorem 4.8 unit ball vector Voln Voln(K zero