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Geometry of Isotropic Convex Bodies
 
Silouanos Brazitikos University of Athens, Athens, Greece
Apostolos Giannopoulos University of Athens, Athens, Greece
Petros Valettas Texas A & M University, College Station, TX
Beatrice-Helen Vritsiou University of Athens, Athens, Greece
Front Cover for Geometry of Isotropic Convex Bodies
Available Formats:
Hardcover ISBN: 978-1-4704-1456-6
Product Code: SURV/196
List Price: $134.00
MAA Member Price: $120.60
AMS Member Price: $107.20
Electronic ISBN: 978-1-4704-1526-6
Product Code: SURV/196.E
List Price: $134.00
MAA Member Price: $120.60
AMS Member Price: $107.20
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $201.00
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Front Cover for Geometry of Isotropic Convex Bodies
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  • Front Cover for Geometry of Isotropic Convex Bodies
  • Back Cover for Geometry of Isotropic Convex Bodies
Geometry of Isotropic Convex Bodies
Silouanos Brazitikos University of Athens, Athens, Greece
Apostolos Giannopoulos University of Athens, Athens, Greece
Petros Valettas Texas A & M University, College Station, TX
Beatrice-Helen Vritsiou University of Athens, Athens, Greece
Available Formats:
Hardcover ISBN:  978-1-4704-1456-6
Product Code:  SURV/196
List Price: $134.00
MAA Member Price: $120.60
AMS Member Price: $107.20
Electronic ISBN:  978-1-4704-1526-6
Product Code:  SURV/196.E
List Price: $134.00
MAA Member Price: $120.60
AMS Member Price: $107.20
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $201.00
MAA Member Price: $180.90
AMS Member Price: $160.80
  • Book Details
     
     
    Mathematical Surveys and Monographs
    Volume: 1962014; 594 pp
    MSC: Primary 52; 46; 60; 28;

    The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension.

    The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

    Readership

    Graduate students and research mathematicians interested in geometric and analytic study of convex bodies.

  • Table of Contents
     
     
    • Chapters
    • Chapter 1. Background from asymptotic convex geometry
    • Chapter 2. Isotropic log-concave measures
    • Chapter 3. Hyperplane conjecture and Bourgain’s upper bound
    • Chapter 4. Partial answers
    • Chapter 5. $L_q$-centroid bodies and concentration of mass
    • Chapter 6. Bodies with maximal isotropic constant
    • Chapter 7. Logarithmic Laplace transform and the isomorphic slicing problem
    • Chapter 8. Tail estimates for linear functionals
    • Chapter 9. $M$ and $M*$-estimates
    • Chapter 10. Approximating the covariance matrix
    • Chapter 11. Random polytopes in isotropic convex bodies
    • Chapter 12. Central limit problem and the thin shell conjecture
    • Chapter 13. The thin shell estimate
    • Chapter 14. Kannan-Lovász-Simonovits conjecture
    • Chapter 15. Infimum convolution inequalities and concentration
    • Chapter 16. Information theory and the hyperplane conjecture
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Volume: 1962014; 594 pp
MSC: Primary 52; 46; 60; 28;

The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension.

The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

Readership

Graduate students and research mathematicians interested in geometric and analytic study of convex bodies.

  • Chapters
  • Chapter 1. Background from asymptotic convex geometry
  • Chapter 2. Isotropic log-concave measures
  • Chapter 3. Hyperplane conjecture and Bourgain’s upper bound
  • Chapter 4. Partial answers
  • Chapter 5. $L_q$-centroid bodies and concentration of mass
  • Chapter 6. Bodies with maximal isotropic constant
  • Chapter 7. Logarithmic Laplace transform and the isomorphic slicing problem
  • Chapter 8. Tail estimates for linear functionals
  • Chapter 9. $M$ and $M*$-estimates
  • Chapter 10. Approximating the covariance matrix
  • Chapter 11. Random polytopes in isotropic convex bodies
  • Chapter 12. Central limit problem and the thin shell conjecture
  • Chapter 13. The thin shell estimate
  • Chapter 14. Kannan-Lovász-Simonovits conjecture
  • Chapter 15. Infimum convolution inequalities and concentration
  • Chapter 16. Information theory and the hyperplane conjecture
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