Geometry Of Isotropic Convex Bodies

Author : Silouanos Brazitikos,Apostolos Giannopoulos,Petros Valettas,Beatrice-Helen Vritsiou
Publisher : American Mathematical Soc.
Page : 618 pages
File Size : 47,7 Mb
Release : 2014-04-24
Category : Mathematics
ISBN : 9781470414566

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Geometry of Isotropic Convex Bodies by Silouanos Brazitikos,Apostolos Giannopoulos,Petros Valettas,Beatrice-Helen Vritsiou Pdf

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.

Author : Maria Moszynska
Publisher : Springer Science & Business Media
Page : 226 pages
File Size : 51,5 Mb
Release : 2006-11-24
Category : Mathematics
ISBN : 9780817644512

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Selected Topics in Convex Geometry by Maria Moszynska Pdf

Examines in detail those topics in convex geometry that are concerned with Euclidean space Enriched by numerous examples, illustrations, and exercises, with a good bibliography and index Requires only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory Can be used for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization

Author : Keith M. Ball,Vitali Milman
Publisher : Cambridge University Press
Page : 260 pages
File Size : 47,9 Mb
Release : 1999-01-28
Category : Mathematics
ISBN : 0521642590

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Convex Geometric Analysis by Keith M. Ball,Vitali Milman Pdf

Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.

Author : Alexander Koldobsky,Vladyslav Yaskin
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 45,8 Mb
Release : 2024-06-14
Category : Mathematics
ISBN : 0821883356

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The Interface Between Convex Geometry and Harmonic Analysis by Alexander Koldobsky,Vladyslav Yaskin Pdf

"The book is written in the form of lectures accessible to graduate students. This approach allows the reader to clearly see the main ideas behind the method, rather than to dwell on technical difficulties. The book also contains discussions of the most recent advances in the subject. The first section of each lecture is a snapshot of that lecture. By reading each of these sections first, novices can gain an overview of the subject, then return to the full text for more details."--BOOK JACKET.

Author : Tommy Bonnesen,Werner Fenchel
Publisher : Unknown
Page : 192 pages
File Size : 54,8 Mb
Release : 1987
Category : Mathematics
ISBN : UOM:39015015605523

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Theory of Convex Bodies by Tommy Bonnesen,Werner Fenchel Pdf

Author : Rolf Schneider
Publisher : Cambridge University Press
Page : 759 pages
File Size : 40,7 Mb
Release : 2014
Category : Mathematics
ISBN : 9781107601017

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Convex Bodies: The Brunn–Minkowski Theory by Rolf Schneider Pdf

A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Author : Bo'az Klartag,Emanuel Milman
Publisher : Springer
Page : 463 pages
File Size : 53,9 Mb
Release : 2014-10-08
Category : Mathematics
ISBN : 9783319094779

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Geometric Aspects of Functional Analysis by Bo'az Klartag,Emanuel Milman Pdf

As in the previous Seminar Notes, the current volume reflects general trends in the study of Geometric Aspects of Functional Analysis. Most of the papers deal with different aspects of Asymptotic Geometric Analysis, understood in a broad sense; many continue the study of geometric and volumetric properties of convex bodies and log-concave measures in high-dimensions and in particular the mean-norm, mean-width, metric entropy, spectral-gap, thin-shell and slicing parameters, with applications to Dvoretzky and Central-Limit-type results. The study of spectral properties of various systems, matrices, operators and potentials is another central theme in this volume. As expected, probabilistic tools play a significant role and probabilistic questions regarding Gaussian noise stability, the Gaussian Free Field and First Passage Percolation are also addressed. The historical connection to the field of Classical Convexity is also well represented with new properties and applications of mixed-volumes. The interplay between the real convex and complex pluri-subharmonic settings continues to manifest itself in several additional articles. All contributions are original research papers and were subject to the usual refereeing standards.

Author : Kurt Leichtweiß
Publisher : Wiley-VCH
Page : 0 pages
File Size : 45,6 Mb
Release : 1999-01-12
Category : Science
ISBN : 3527402616

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Affine Geometry of Convex Bodies by Kurt Leichtweiß Pdf

The theory of convex bodies is nowadays an important independent topic of geometry. The author discusses the equiaffine geometry and differential geometry of convex bodies and convex surfaces and especially stresses analogies to classical Euclidean differential geometry. These theories are illustrated by practical applications in areas such as shipbuilding. He offers an accessible introduction to the latest developments in the subject.

Author : Alexander Koldobsky
Publisher : American Mathematical Soc.
Page : 170 pages
File Size : 42,5 Mb
Release : 2014-11-12
Category : Electronic
ISBN : 9781470419523

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Fourier Analysis in Convex Geometry by Alexander Koldobsky Pdf

The study of the geometry of convex bodies based on information about sections and projections of these bodies has important applications in many areas of mathematics and science. In this book, a new Fourier analysis approach is discussed. The idea is to express certain geometric properties of bodies in terms of Fourier analysis and to use harmonic analysis methods to solve geometric problems. One of the results discussed in the book is Ball's theorem, establishing the exact upper bound for the -dimensional volume of hyperplane sections of the -dimensional unit cube (it is for each ). Another is the Busemann-Petty problem: if and are two convex origin-symmetric -dimensional bodies and the -dimensional volume of each central hyperplane section of is less than the -dimensional volume of the corresponding section of , is it true that the -dimensional volume of is less than the volume of ? (The answer is positive for and negative for .) The book is suitable for graduate students and researchers interested in geometry, harmonic and functional analysis, and probability. Prerequisites for reading this book include basic real, complex, and functional analysis.

Author : Horst Martini,Luis Montejano,Déborah Oliveros
Publisher : Springer
Page : 486 pages
File Size : 54,6 Mb
Release : 2019-03-16
Category : Mathematics
ISBN : 9783030038687

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Bodies of Constant Width by Horst Martini,Luis Montejano,Déborah Oliveros Pdf

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts. An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields) Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces) The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods) Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.) Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics) The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.) Technical applications, such as film projectors, the square-hole drill, and rotary engines Bodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.

Author : K. Leichtweiss
Publisher : Unknown
Page : 310 pages
File Size : 54,6 Mb
Release : 1998-01-01
Category : Convex bodies
ISBN : 3335005147

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Affine Geometry of Convex Bodies by K. Leichtweiss Pdf

Author : Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman
Publisher : American Mathematical Soc.
Page : 451 pages
File Size : 51,9 Mb
Release : 2015-06-18
Category : Functional analysis
ISBN : 9781470421939

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Asymptotic Geometric Analysis, Part I by Shiri Artstein-Avidan, Apostolos Giannopoulos, Vitali D. Milman Pdf

The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomenon", one of the most powerful tools of the theory, responsible for many counterintuitive results. A central theme in this book is the interaction of randomness and pattern. At first glance, life in high dimension seems to mean the existence of multiple "possibilities", so one may expect an increase in the diversity and complexity as dimension increases. However, the concentration of measure and effects caused by convexity show that this diversity is compensated and order and patterns are created for arbitrary convex bodies in the mixture caused by high dimensionality. The book is intended for graduate students and researchers who want to learn about this exciting subject. Among the topics covered in the book are convexity, concentration phenomena, covering numbers, Dvoretzky-type theorems, volume distribution in convex bodies, and more.

Author : Paolo Ciatti,Alessio Martini
Publisher : Springer Nature
Page : 488 pages
File Size : 40,9 Mb
Release : 2021-09-27
Category : Mathematics
ISBN : 9783030720582

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Geometric Aspects of Harmonic Analysis by Paolo Ciatti,Alessio Martini Pdf

This volume originated in talks given in Cortona at the conference "Geometric aspects of harmonic analysis" held in honor of the 70th birthday of Fulvio Ricci. It presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest mathematicians working in these areas. The subjects dealt with are topics of current interest in closely interrelated areas of Fourier analysis, singular integral operators, oscillatory integral operators, partial differential equations, multilinear harmonic analysis, and several complex variables. The work is addressed to researchers in the field.

Author : David Alonso-Gutiérrez,Jesús Bastero
Publisher : Springer
Page : 148 pages
File Size : 54,7 Mb
Release : 2015-01-07
Category : Mathematics
ISBN : 9783319132631

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Approaching the Kannan-Lovász-Simonovits and Variance Conjectures by David Alonso-Gutiérrez,Jesús Bastero Pdf

Focusing on two central conjectures of Asymptotic Geometric Analysis, the Kannan-Lovász-Simonovits spectral gap conjecture and the variance conjecture, these Lecture Notes present the theory in an accessible way, so that interested readers, even those who are not experts in the field, will be able to appreciate the treated topics. Offering a presentation suitable for professionals with little background in analysis, geometry or probability, the work goes directly to the connection between isoperimetric-type inequalities and functional inequalities, giving the interested reader rapid access to the core of these conjectures. In addition, four recent and important results in this theory are presented in a compelling way. The first two are theorems due to Eldan-Klartag and Ball-Nguyen, relating the variance and the KLS conjectures, respectively, to the hyperplane conjecture. Next, the main ideas needed prove the best known estimate for the thin-shell width given by Guédon-Milman and an approach to Eldan's work on the connection between the thin-shell width and the KLS conjecture are detailed.

Author : Artur Avila,Michael Th. Rassias,Yakov Sinai
Publisher : Springer Nature
Page : 388 pages
File Size : 50,8 Mb
Release : 2022-11-01
Category : Mathematics
ISBN : 9783031053313

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Analysis at Large by Artur Avila,Michael Th. Rassias,Yakov Sinai Pdf

​Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgain’s discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prékopa-Leindler inequality and sumsets with quasicubes, the fractal uncertainty principle for the Walsh-Fourier transform, the continuous formulation of shallow neural networks as Wasserstein-type gradient flows, logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrödinger operators, polynomial equations in subgroups, trace sets of restricted continued fraction semigroups, exponential sums, twisted multiplicativity and moments, the ternary Goldbach problem, as well as the multiplicative group generated by two primes in Z/QZ. It is hoped that this volume will inspire further research in the areas of analysis treated in this book and also provide direction and guidance for upcoming developments in this essential subject of mathematics.