## 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 upperbound 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 centralhyperplane 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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### Contents

Introduction | 1 |

Basic Concepts | 13 |

Volume and the Fourier Transform | 49 |

Intersection Bodies | 71 |

The BusemannPetty Problem | 95 |

Intersection Bodies and LpSpaces | 116 |

Extremal Sections of Balls | 143 |

Projections and the Fourier Transform | 151 |

163 | |

Index | 169 |

### Common terms and phrases

affirmative argument body in Rn Brunn,s theorem Busemann Busemann-Petty problem central hyperplane section constant continuous function converges cube defined dimension embedding in Lp embeds in Lp embeds isometrically Euclidean ball exists a finite fact fc-intersection body finite Borel measure Fourier transform Fubini theorem function exp function F function of degree homogeneous function homogeneous of degree hyperplane sections inequality integrable function intersection body isometrically in Lp Lemma locally integrable Lp-spaces Lutwak Minkowski functional n-dimensional negative non-negative normed space origin 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 random variable result follows right-hand side second derivative solution sphere Sn~l spherical harmonics spherical Radon transform subspace of Lq Suppose surface area measure symmetric convex body symmetric star body test function Theorem 3.8 unit ball vector Voln(L ydv(y zero