تحلیل محدب: نظریه و کاربردها (ترجمه‌های مونوگراف‌های ریاضی)

Convex analysis is a branch of mathematics that studies convex sets, convex functions, and convex extremal problems. It has surprisingly diverse and fruitful applications in mathematics, mathematical physics, technology, and economics. This book is an introduction to convex analysis and some of its applications. It starts with basic theory, which is explained within the framework of finite-dimensional spaces. The only prerequisites are basic analysis and simple geometry. The second chapter presents some applications of convex analysis, including problems of linear programming, geometry, and approximation. Special attention is paid to applications of convex analysis to Kolmogorov-type inequalities for derivatives of functions in one variable. Chapter 3 collects some results on geometry and convex analysis in infinite-dimensional spaces. A comprehensive introduction written “for beginners” illustrates the fundamentals of convex analysis in finite-dimensional spaces. The book can be used for an advanced undergraduate or graduate-level course on convex analysis and its applications. It is also suitable for independent study of this extremely important area of mathematics. "This book is an introduction to convex analysis and some of its applications. It starts with basis theory, which is explained within the framework of finite-dimensional spaces. The only prerequisites are basic analysis and simple geometry. The second chapter presents some applications of convex analysis, including problems of linear programming, geometry, and approximation. Special attention is paid to applications of convex analysis to Kolmogorov-type inequalities for derivatives of functions is one variable. Chapter 3 collects some results on geometry and convex analysis in infinite-dimensional spaces. A comprehensive introduction written "for beginners" illustrates the fundamentals of convex analysis in finite-dimensional spaces." "The book can be used for an advanced undergraduate or graduate level course on convex analysis and its applications. It is also suitable for independent study of this extremely important area of mathematics."--Jacket Ch. 1. Theory -- 1. Basic Definitions -- 2. Duality In Convex Analysis -- 3. Convex Calculus -- 4. Finite-dimensional Convex Geometry -- 5. Convex Extremal Problems -- 6. Supplement: Convex Analysis In Vector Spaces -- Ch. 2. Applications -- 7. Convex Analysis Of Subspaces And Cones And The Theory Of Linear Equations And Inequalities -- 8. Classical Inequalities, Problems Of Geometry And Mechanics -- 9. Kolmogorov-type Inequalities For Derivatives -- 10. Convex Analysis And Extremal Problems Of Approximation And Recovery -- Ch. 3. Appendix -- 11. Basic Theorems Of Convex Analysis -- 12. Supplementary Topics Of Convex Analysis -- 13. Convex Analysis And The Theory Of Extremum. G.g. Magaril-ilʹyaev, V.m. Tikhomirov ; Translated By Dmitry Chibisov. Includes Bibliographical References (p. 177-180) And Index. This graduate textbook illustrates the fundamentals of convex analysis in a finite dimensional setting, examines the properties of convex sets and convex functions, and provides applications of convexity theory to the duality of convex calculus and to extremal problems of approximation and recovery. Originally published in Russian as Vypuklyi analiz teoriia i prilozheniia in 2000 Presents an introduction to convex analysis and some of its applications. This work starts with basic theory, which is explained within the framework of finite-dimensional spaces. It presents some applications of convex analysis, including problems of linear programming, geometry, and approximation.