Fourier analysis in convex geometry / Alexander Koldobsky.

Format:

Book

Author/Creator:

Koldobsky, Alexander, 1955- author.

Series:

Mathematical surveys and monographs ; no. 116.

Mathematical surveys and monographs, 0076-5376 ; volume 116

Language:

English

Subjects (All):

Convex sets.

Banach spaces.

Fourier transformations.

Physical Description:

1 online resource (178 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2005]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""Chapter 2. Basic Concepts""; ""2.1. Star bodies""; ""2.2. Convex bodies""; ""2.3. Radon transforms""; ""2.4. The Gamma-function""; ""2.5. The Fourier transform of distributions""; ""2.6. Fractional derivatives""; ""2.7. Positive definite distributions""; ""2.8. Stable random variables and the function γ[sub(q)]""; ""Chapter 3. Volume and the Fourier Transform""; ""3.1. The first examples: hyperplane sections of l[sub(q)]-balls""; ""3.2. A general formula for the volume of hyperplane sections""

""3.3. The parallel section function and the Fourier transform""""3.4. Parseval's formula on the sphere""; ""3.5. Remarks and further results""; ""Chapter 4. Intersection Bodies""; ""4.1. A Fourier analytic characterization""; ""4.2. k-intersection bodies""; ""4.3. L[sub(p)-balls as k-intersection bodies""; ""4.4. The second derivative test""; ""4.5. Remarks and further results""; ""Chapter 5. The Busemann-Petty Problem""; ""5.1. A Fourier analytic solution""; ""5.2. How can one make the answer affirmative?""; ""5.3. The affirmative part via spherical harmonics""

""5.4. Zvavitch's generalization to arbitrary measures""""5.5. Remarks and further results""; ""Chapter 6. Intersection Bodies and L[sub(p)]-Spaces""; ""6.1. L[sub(p)]-spaces and positive definite functions""; ""6.2. Schoenberg's problems on positive definite functions""; ""6.3. Intersection bodies and embeddings in L[sub(p)], p < 0""; ""6.4. Remarks and further results""; ""Chapter 7. Extremal Sections of l[sub(q)]-Balls""; ""7.1. The case of the cube, K. Ball's theorem""; ""7.2. The case . . .""; ""7.3. Remarks and further results""; ""Chapter 8. Projections and the Fourier Transform""

""8.1. A formula for the volume of hyperplane projections""""8.2. Extremal hyperplane projections of l[sub(q)]-balls""; ""8.3. Projection bodies""; ""8.4. The Shephard problem""; ""8.5. Remarks and further results""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 163-168) and index.

Description based on print version record.

ISBN:

1-4704-1343-4