Geometry of Isotropic Convex Bodies

Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, Beatrice-Helen Vritsiou (Autoren)

Buch | Hardcover

594 Seiten

2014
American Mathematical Society (Verlag)
978-1-4704-1456-6 (ISBN)

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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 normalisation, 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-Lovasz-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.

Silouanos Brazitikos, University of Athens, Greece. Apostolos Giannopoulos, University of Athens, Greece. Petros Valettas, Texas A & M University, College Station, TX. Beatrice-Helen Vritsiou, University of Athens, Greece.

Background from asymptotic convex geometry
Isotropic log-concave measures
Hyperplane conjecture and Bourgain's upper bound
Partial answers
L q -centroid bodies and concentration of mass
Bodies with maximal isotropic constant
Logarithmic Laplace transform and the isomorphic slicing problem
Tail estimates for linear functionals
M and M? *-estimates
Approximating the covariance matrix
Random polytopes in isotropic convex bodies
Central limit problem and the thin shell conjecture
The thin shell estimate
Kannan-Lovász-Simonovits conjecture
Infimum convolution inequalities and concentration
Information theory and the hyperplane conjecture
Bibliography
Subject index
Author index

Erscheint lt. Verlag	30.5.2014
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	1225 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-1456-2 / 1470414562
ISBN-13	978-1-4704-1456-6 / 9781470414566
Zustand	Neuware