目录
 前辅文
 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 γq
 Chapter 3. Volume and the Fourier Transform
  3.1. The first examples: hyperplane sections of ℓq-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. Lp-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 Lp-Spaces
  6.1. Lp-spaces and positive definite functions
  6.2. Schoenberg's problems on positive definite functions
  6.3. Intersection bodies and embeddings in Lp, p < 0
  6.4. Remarks and further results
 Chapter 7. Extremal Sections of ℓq-Balls
  7.1. The case of the cube, K. Ball's theorem
  7.2. The case 0   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 ℓq-balls
  8.3. Projection bodies
  8.4. The Shephard problem
  8.5. Remarks and further results
 Bibliography
 Index
 前辅文
 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 γq
 Chapter 3. Volume and the Fourier Transform
  3.1. The first examples: hyperplane sections of ℓq-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. Lp-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 Lp-Spaces
  6.1. Lp-spaces and positive definite functions
  6.2. Schoenberg's problems on positive definite functions
  6.3. Intersection bodies and embeddings in Lp, p < 0
  6.4. Remarks and further results
 Chapter 7. Extremal Sections of ℓq-Balls
  7.1. The case of the cube, K. Ball's theorem
  7.2. The case 0   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 ℓq-balls
  8.3. Projection bodies
  8.4. The Shephard problem
  8.5. Remarks and further results
 Bibliography
 Index