This volume presents a rigorous study of the Brunn–Minkowski theory, the isoperimetric inequality, and their functional and analytic extensions. It develops the classical theory of convex bodies and mixed volumes, highlighting the geometric foundations of volume and surface-area inequalities. Building on this, the text examines isoperimetric problems in both the classical and measure-theoretic settings, including sets of finite perimeter, and establishes functional analogues of geometric inequalities. A significant part of the book is devoted to the interaction between these inequalities and curvature-related equations, with Monge–Ampère equations on the sphere serving as a principal example. The exposition emphasizes structural relations: how classical convex geometry informs functional inequalities, and how analytic methods extend geometric results to broader settings. Additional topics include symmetrization techniques, stability issues, and connections with variational principles, illustrating the conceptual bridges between geometry, analysis, and partial differential equations. Throughout, the presentation integrates classical results, modern extensions, and analytic tools to provide a coherent framework for understanding the interplay between volume, surface, curvature, and functional inequalities.
Download pp. xi–xv
Basic notions and notation
pp. 1–4
1 Preliminaries in convex geometry in R
n
pp. 5–60
2 Surface area, surface area measure and cone volume measure for convex bodies in R
n
pp. 61–92
3 The Brunn–Minkowski and the Prékopa–Leindler inequalities in the measurable case
pp. 93–120
4 The isoperimetric inequality in the case of Lipschitz boundary
pp. 121–176
5 The isoperimetric inequality for sets of finite perimeter in R
n
and the Sobolev inequality for BV functions
pp. 177–202
6 Associated ellipsoids, Blaschke–Santaló inequality and the reverse isoperimetric inequality
pp. 203–260
7 Steiner formula and mixed volumes
pp. 261–302
8 Convex bodies and Gaussian curvature
pp. 303–390
9 The Minkowski problem, the L
p
-Minkowski problem, and the L
p
-Brunn–Minkowski inequality/conjecture
pp. 391–454
A Appendix: Background from analysis and algebra
pp. 455–480
References
pp. 481–519
Index
pp. 521–523
*
Pages: 288
ISBN: 978-1-80061-859-6 (hardcover)
ISBN: 978-1-80061-877-0 (softcover)
This book offers a clear and comprehensive introduction to the Lebesgue integral — one of the foundational concepts of modern analysis. Beginning with the historical development of integration, it builds naturally from the notions of null sets and step functions toward more advanced topics such as measurable functions, convergence theorems, Fubini's theorem, change of variables, and the structure of Lp spaces. Throughout, the material is presented with a focus on clarity, logical progression, and practical insight.
Spanning eight chapters, the book guides readers through both the theoretical foundations and practical applications of the Lebesgue integral in ℝN. Along the way, it explores a wide range of key ideas, including the characterization of Riemann integrability, the Tonelli–Hobson criterion, non-measurable sets, integral transformations, Cavalieri's principle, Eulerian integrals, and convolution of functions. The result is a well-rounded and accessible treatment that bridges classical calculus with the depth of real analysis.
Each chapter concludes with a carefully selected set of problems, all of which are fully solved in a dedicated section — making this an ideal resource for both independent study and structured coursework. Whether you are encountering measure theory for the first time or seeking a deeper understanding of integration in higher dimensions, this book offers the theoretical foundation and practical support needed to master one of mathematics' most powerful analytical tools.
Preface
About the Author
Theory:
Historical Notes
The Lebesgue Integral in ℝN
The Convergence Theorems
Fubini's Theorem
Functions and Measurables Sets
Transformation of Integrals
Some Calculus Techniques
The Lp Spaces
Solutions:
Solutions to the Problems
Bibliography
Index
This textbook is designed for undergraduate and graduate students pursuing degrees in Mathematics, Physics, or Engineering, providing a clear and rigorous introduction to advanced integral concepts essential for these fields.
This book explores the concept of maxima and minima (extreme values) throughout mathematics. While extrema are commonly associated with calculus, the authors demonstrate that the idea of finding optimal values is fundamental across many mathematical disciplines, including geometry, arithmetic, algebra, and probability.
The book is divided into two major parts. The first part introduces the "method of extremes" and illustrates how maximum and minimum principles can be applied to solve practical and mathematical problems. The second and larger part investigates extreme-value problems without using derivatives, emphasizing logical reasoning, geometric arguments, and elementary methods.
A particular focus is placed on geometry, where optimization problems often reveal elegant and surprising properties of figures and constructions. The authors aim to deepen mathematical understanding while providing teachers and students with powerful problem-solving techniques.
A complete chapter-by-chapter table of contents was not publicly available at the time of the search. Based on the publisher's description, the book covers the following themes:
Main Topics Include
Note: The publisher has released a detailed description of the book, but the full official table of contents does not yet appear to be publicly available. The contents above are reconstructed from the available publisher metadata and description.
Asymptotic Analysis is a comprehensive graduate-level text and reference work devoted to modern asymptotic methods and their applications. The volume consists of 30 chapters organized into five parts, providing both classical foundations and recent developments in the field.
The book covers a broad range of topics, including:
The authors emphasize both rigorous mathematical theory and practical applications. The text demonstrates how asymptotic methods are used in:
The book is intended for advanced undergraduate students, graduate students, researchers, and instructors. Exercises are included to support self-study and classroom use.
The publisher's complete chapter-by-chapter table of contents has not yet been publicly released. However, the book is described as containing 30 chapters in five parts covering the following major themes:
Part I. Foundations of Asymptotic Analysis
Part II. Special Functions and Orthogonal Polynomials
Part III. Integrals and Approximation Methods
Part IV. Differential and Difference Equations
Part V. Modern Developments and Applications
Note: The above contents are reconstructed from the official publisher's description. A detailed official chapter list has not yet been made publicly available