Fourier Analysis In Convex Geometry (Mathematical Surveys and Monographs)

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. 0PP1......Page 1 1......Page 2 2......Page 3 3......Page 4 4......Page 5 5......Page 6 6......Page 7 7......Page 8 8PP10......Page 9 PA001......Page 10 PA002......Page 11 PA003......Page 12 PA004......Page 13 PA005......Page 14 PA006......Page 15 PA007......Page 16 PA008......Page 17 PA009......Page 18 PA010......Page 19 PA011......Page 20 PA012......Page 21 PA013......Page 22 PA014......Page 23 PA015......Page 24 PA016......Page 25 PA017......Page 26 PA018......Page 27 PA019......Page 28 PA020......Page 29 PA021......Page 30 PA022......Page 31 PA023......Page 32 PA024......Page 33 PA025......Page 34 PA026......Page 35 PA027......Page 36 PA028......Page 37 PA029......Page 38 PA030......Page 39 PA031......Page 40 PA032......Page 41 PA033......Page 42 PA034......Page 43 PA035......Page 44 PA036......Page 45 PA037......Page 46 PA038......Page 47 PA039......Page 48 PA040......Page 49 PA041......Page 50 PA042......Page 51 PA043......Page 52 PA044......Page 53 PA045......Page 54 PA046......Page 55 PA047......Page 56 PA048......Page 57 PA049......Page 58 PA050......Page 59 PA051......Page 60 PA052......Page 61 PA053......Page 62 PA054......Page 63 PA055......Page 64 PA056......Page 65 PA057......Page 66 PA058......Page 67 PA059......Page 68 PA060......Page 69 PA061......Page 70 PA062......Page 71 PA063......Page 72 PA064......Page 73 PA065......Page 74 PA066......Page 75 PA067......Page 76 PA068......Page 77 PA069......Page 78 PA070......Page 79 PA071......Page 80 PA072......Page 81 PA073......Page 82 PA074......Page 83 PA075......Page 84 PA076......Page 85 PA077......Page 86 PA078......Page 87 PA079......Page 88 PA080......Page 89 PA081......Page 90 PA082......Page 91 PA082a......Page 92 PA083......Page 93 PA083a......Page 94 PA084......Page 95 PA085......Page 96 PA086......Page 97 PA087......Page 98 PA088......Page 99 PA089......Page 100 PA090......Page 101 PA091......Page 102 PA091a......Page 103 PA092......Page 104 PA092a......Page 105 PA093......Page 106 PA093a......Page 107 PA094......Page 108 PA100......Page 109 PA101......Page 110 PA102......Page 111 PA103......Page 112 PA104......Page 113 PA105......Page 114 PA106......Page 115 PA107......Page 116 PA108......Page 117 PA109......Page 118 PA110......Page 119 PA111......Page 120 PA112......Page 121 PA113......Page 122 PA114......Page 123 PA115......Page 124 PA116......Page 125 PA117......Page 126 PA118......Page 127 PA119......Page 128 PA120......Page 129 PA121......Page 130 PA122......Page 131 PA123......Page 132 PA124......Page 133 PA125......Page 134 PA126......Page 135 PA127......Page 136 PA128......Page 137 PA129......Page 138 PA130......Page 139 PA131......Page 140 PA132......Page 141 PA133......Page 142 PA134......Page 143 PA135......Page 144 PA136......Page 145 PA137......Page 146 PA138......Page 147 PA139......Page 148 PA140......Page 149 PA141......Page 150 PA142......Page 151 PA143......Page 152 PA144......Page 153 PA145......Page 154 PA146......Page 155 PA147......Page 156 PA148......Page 157 PA149......Page 158 PA150......Page 159 PA151......Page 160 PA152......Page 161 PA153......Page 162 PA154......Page 163 PA155......Page 164 PA156......Page 165 PA157......Page 166 PA158......Page 167 PA159......Page 168 PA160......Page 169 PA161......Page 170 PA162......Page 171 PA163......Page 172 PA164......Page 173 PA165......Page 174 PA166......Page 175 PA167......Page 176 Pa168......Page 177 PA169......Page 178 PA170......Page 179 PT1......Page 180

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.

"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." "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."--Jacket 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.