The Brunn-Minkowski Inequality and Related Results

The Brunn-Minkowski Inequality and Related Results
Title The Brunn-Minkowski Inequality and Related Results PDF eBook
Author Trista A. Mullin
Publisher
Pages 0
Release 2018
Genre
ISBN

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The Brunn-Minkowski Inequality is a classical result that compares the volumes of twosets, in particular convex bodies, and the volume of their Minkowski sum. The proof iselegant and the eects are far reaching in mathematics. In this thesis we will examinethe proof of the inequality, and its multiplicative and integral forms. From there wewill explore a few applications and an analog to Brunn's slice theorem. Additionally, wewill look at how the Brunn-Minkowski Inequality can be used to obtain results regardinggeneral log concave measures, isoperimetric inequalities, and spherical concentrations.We will end the journey with a quick look at what can be said about the intersectionbody of a convex body.

Theory of Convex Bodies

Theory of Convex Bodies
Title Theory of Convex Bodies PDF eBook
Author Tommy Bonnesen
Publisher
Pages 192
Release 1987
Genre Mathematics
ISBN

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Convex Bodies: The Brunn–Minkowski Theory

Convex Bodies: The Brunn–Minkowski Theory
Title Convex Bodies: The Brunn–Minkowski Theory PDF eBook
Author Rolf Schneider
Publisher Cambridge University Press
Pages 759
Release 2014
Genre Mathematics
ISBN 1107601010

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

Lectures on Convex Geometry

Lectures on Convex Geometry
Title Lectures on Convex Geometry PDF eBook
Author Daniel Hug
Publisher Springer Nature
Pages 287
Release 2020-08-27
Genre Mathematics
ISBN 3030501809

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This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Asymptotic Geometric Analysis, Part I

Asymptotic Geometric Analysis, Part I
Title Asymptotic Geometric Analysis, Part I PDF eBook
Author Shiri Artstein-Avidan
Publisher American Mathematical Soc.
Pages 473
Release 2015-06-18
Genre Mathematics
ISBN 1470421933

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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.

Inequalities

Inequalities
Title Inequalities PDF eBook
Author Elliott H. Lieb
Publisher Springer Science & Business Media
Pages 687
Release 2012-12-06
Genre Mathematics
ISBN 3642559255

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Inequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. E. H. Lieb has discovered a host of inequalities that are enormously useful in mathematics as well as in physics. His results are collected in this book which should become a standard source for further research. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics.

Geometric Tomography

Geometric Tomography
Title Geometric Tomography PDF eBook
Author Richard J. Gardner
Publisher Cambridge University Press
Pages 7
Release 2006-06-19
Genre Mathematics
ISBN 0521866804

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Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. It overlaps with convex geometry, and employs many tools from that area including integral geometry. It also has connections to geometric probing in robotics and to stereology. The main text contains a rigorous treatment of the subject starting from basic concepts and moving up to the research frontier: seventy-two unsolved problems are stated. Each chapter ends with extensive notes, historical remarks, and some biographies. This comprehensive work will be invaluable to specialists in geometry and tomography; the opening chapters can also be read by advanced undergraduate students.