The Brunn-Minkowski Inequality and Related Results PDF Download
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Author: Trista A. Mullin
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
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.
Author: Trista A. Mullin
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
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.
Author: Tommy Bonnesen
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 192
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Book Description
Author: Rolf Schneider
Publisher: Cambridge University Press
ISBN: 1107601010
Category : Mathematics
Languages : en
Pages : 759
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Book Description
A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.
Author: Daniel Hug
Publisher: Springer Nature
ISBN: 3030501809
Category : Mathematics
Languages : en
Pages : 287
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Book Description
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.
Author: Shiri Artstein-Avidan
Publisher: American Mathematical Soc.
ISBN: 1470421933
Category : Mathematics
Languages : en
Pages : 473
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Book Description
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: Elliott H. Lieb
Publisher: Springer Science & Business Media
ISBN: 3642559255
Category : Mathematics
Languages : en
Pages : 687
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Book Description
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.
Author: Richard J. Gardner
Publisher: Cambridge University Press
ISBN: 0521866804
Category : Mathematics
Languages : en
Pages : 7
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Book Description
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.
Author: Gilles Pisier
Publisher: Cambridge University Press
ISBN: 9780521666350
Category : Mathematics
Languages : en
Pages : 270
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Book Description
A self-contained presentation of results relating the volume of convex bodies and Banach space geometry.
Author: Roman Vershynin
Publisher: Cambridge University Press
ISBN: 1108244548
Category : Mathematics
Languages : en
Pages : 299
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Book Description
High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions. Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more. It is the first to integrate theory, key tools, and modern applications of high-dimensional probability. Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality. It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension. A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression.
Author: P.S. Bullen
Publisher: Springer Science & Business Media
ISBN: 9401722269
Category : Mathematics
Languages : en
Pages : 476
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Book Description
Approach your problems from the right end It isn't !hat they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal 0/ Fa/her 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin'. van GuJik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fie1ds does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "complete1y integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing c1assification schemes. They draw upon wide1y different sections of mathematics.
