Author : Keith M. Ball
Publisher : Cambridge University Press
Page : 260 pages
File Size : 19,98 MB
Release : 1999-01-28
Category : Mathematics
ISBN : 9780521642590
Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.
Author : Daniel Hug
Publisher : Springer Nature
Page : 287 pages
File Size : 41,79 MB
Release : 2020-08-27
Category : Mathematics
ISBN : 3030501809
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 : Alexander Koldobsky
Publisher : American Mathematical Soc.
Page : 178 pages
File Size : 16,37 MB
Release : 2014-11-12
Category : Mathematics
ISBN : 1470419521
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 : Steven G. Krantz
Publisher : CRC Press
Page : 174 pages
File Size : 33,58 MB
Release : 2014-10-20
Category : Mathematics
ISBN : 149870638X
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.Convex Analysis introduces
Author : Ilya J. Bakelman
Publisher : Springer Science & Business Media
Page : 524 pages
File Size : 45,80 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 3642698816
Investigations in modem nonlinear analysis rely on ideas, methods and prob lems from various fields of mathematics, mechanics, physics and other applied sciences. In the second half of the twentieth century many prominent, ex emplary problems in nonlinear analysis were subject to intensive study and examination. The united ideas and methods of differential geometry, topology, differential equations and functional analysis as well as other areas of research in mathematics were successfully applied towards the complete solution of com plex problems in nonlinear analysis. It is not possible to encompass in the scope of one book all concepts, ideas, methods and results related to nonlinear analysis. Therefore, we shall restrict ourselves in this monograph to nonlinear elliptic boundary value problems as well as global geometric problems. In order that we may examine these prob lems, we are provided with a fundamental vehicle: The theory of convex bodies and hypersurfaces. In this book we systematically present a series of centrally significant results obtained in the second half of the twentieth century up to the present time. Particular attention is given to profound interconnections between various divisions in nonlinear analysis. The theory of convex functions and bodies plays a crucial role because the ellipticity of differential equations is closely connected with the local and global convexity properties of their solutions. Therefore it is necessary to have a sufficiently large amount of material devoted to the theory of convex bodies and functions and their connections with partial differential equations.
Author : Paul J. Kelly
Publisher :
Page : 0 pages
File Size : 17,86 MB
Release : 2009
Category : Convex bodies
ISBN : 9780486469805
This text assumes no prerequisites, offering an easy-to-read treatment with simple notation and clear, complete proofs. From motivation to definition, its explanations feature concrete examples and theorems. 1979 edition.
Author : Shiri Artstein-Avidan
Publisher : American Mathematical Soc.
Page : 473 pages
File Size : 50,25 MB
Release : 2015-06-18
Category : Mathematics
ISBN : 1470421933
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 : Bozzano G Luisa
Publisher : Elsevier
Page : 769 pages
File Size : 17,5 MB
Release : 2014-06-28
Category : Mathematics
ISBN : 0080934404
Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.
Author : Peter M. Gruber
Publisher : Springer Science & Business Media
Page : 590 pages
File Size : 19,47 MB
Release : 2007-05-17
Category : Mathematics
ISBN : 3540711333
Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other subdisciplines. This book provides a comprehensive overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers, and useful to people working in the applied fields.
Author : Grigoriy Blekherman
Publisher : SIAM
Page : 487 pages
File Size : 18,34 MB
Release : 2013-03-21
Category : Mathematics
ISBN : 1611972280
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
