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| Author | : Keith M. Ball |
| Publisher | : Cambridge University Press |
| Total Pages | : 260 |
| Release | : 1999-01-28 |
| Genre | : 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 |
| Total Pages | : 287 |
| Release | : 2020-08-27 |
| Genre | : 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 | : Steven G. Krantz |
| Publisher | : CRC Press |
| Total Pages | : 174 |
| Release | : 2014-10-20 |
| Genre | : 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 | : Alexander Koldobsky |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 178 |
| Release | : 2014-11-12 |
| Genre | : 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 | : Ilya J. Bakelman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 524 |
| Release | : 2012-12-06 |
| Genre | : 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 | : Shiri Artstein-Avidan |
| Publisher | : American Mathematical Society |
| Total Pages | : 645 |
| Release | : 2021-12-13 |
| Genre | : Mathematics |
| ISBN | : 1470463601 |
This book is a continuation of Asymptotic Geometric Analysis, Part I, which was published as volume 202 in this series. Asymptotic geometric analysis studies properties of geometric objects, such as normed spaces, convex bodies, or convex functions, when the dimensions of these objects increase to infinity. The asymptotic approach reveals many very novel phenomena which influence other fields in mathematics, especially where a large data set is of main concern, or a number of parameters which becomes uncontrollably large. One of the important features of this new theory is in developing tools which allow studying high parametric families. Among the topics covered in the book are measure concentration, isoperimetric constants of log-concave measures, thin-shell estimates, stochastic localization, the geometry of Gaussian measures, volume inequalities for convex bodies, local theory of Banach spaces, type and cotype, the Banach-Mazur compactum, symmetrizations, restricted invertibility, and functional versions of geometric notions and inequalities.
| Author | : Maria Moszynska |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 223 |
| Release | : 2006-11-24 |
| Genre | : Mathematics |
| ISBN | : 0817644512 |
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 | : Paul J. Kelly |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2009 |
| Genre | : 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 | : Silouanos Brazitikos |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 618 |
| Release | : 2014-04-24 |
| Genre | : Mathematics |
| ISBN | : 1470414562 |
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 | : Bozzano G Luisa |
| Publisher | : Elsevier |
| Total Pages | : 769 |
| Release | : 2014-06-28 |
| Genre | : 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.
