Hardcover ISBN:  9781470414566 
Product Code:  SURV/196 
List Price:  $134.00 
MAA Member Price:  $120.60 
AMS Member Price:  $107.20 
Electronic ISBN:  9781470415266 
Product Code:  SURV/196.E 
List Price:  $134.00 
MAA Member Price:  $120.60 
AMS Member Price:  $107.20 

Book DetailsMathematical Surveys and MonographsVolume: 196; 2014; 594 ppMSC: Primary 52; 46; 60; 28;
The study of highdimensional 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 highdimensional 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 wellknown questions regarding the distribution of volume in highdimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the KannanLovászSimonovits conjecture. This book provides a selfcontained and up to date account of the progress that has been made in the last fifteen years.ReadershipGraduate 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 logconcave 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. KannanLovászSimonovits conjecture

Chapter 15. Infimum convolution inequalities and concentration

Chapter 16. Information theory and the hyperplane conjecture


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The study of highdimensional 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 highdimensional 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 wellknown questions regarding the distribution of volume in highdimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the KannanLovászSimonovits conjecture. This book provides a selfcontained and up to date account of the progress that has been made in the last fifteen years.
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 logconcave 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. KannanLovászSimonovits conjecture

Chapter 15. Infimum convolution inequalities and concentration

Chapter 16. Information theory and the hyperplane conjecture