Geometry of Isotropic Convex Bodies
Silouanos Brazitikos · Apostolos Giannopoulos · Petros Valettas · Beatrice-Helen Vritsiou
apr 2014 · Mathematical Surveys and Monographs Libro 196 · American Mathematical Soc.
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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 normalization, the
answer to many fundamental questions should be independent of the
dimension.
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.
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