Kooperativer Bibliotheksverbund

UID:

Format: XIX, 1184 p. 86 illus., 13 illus. in color. , online resource.

Edition: 1st ed. 2024.

ISBN: 9783031505072

Series Statement: Cornerstones,

Content: This text gives a comprehensive introduction to the "common core" of convex geometry. Basic concepts and tools which are present in all branches of that field are presented with a highly didactic approach. Mainly directed to graduate and advanced undergraduates, the book is self-contained in such a way that it can be read by anyone who has standard undergraduate knowledge of analysis and of linear algebra. Additionally, it can be used as a single reference for a complete introduction to convex geometry, and the content coverage is sufficiently broad that the reader may gain a glimpse of the entire breadth of the field and various subfields. The book is suitable as a primary text for courses in convex geometry and also in discrete geometry (including polytopes). It is also appropriate for survey type courses in Banach space theory, convex analysis, differential geometry, and applications of measure theory. Solutions to all exercises are available to instructors who adopt the text for coursework. Most chapters use the same structure with the first part presenting theory and the next containing a healthy range of exercises. Some of the exercises may even be considered as short introductions to ideas which are not covered in the theory portion. Each chapter has a notes section offering a rich narrative to accompany the theory, illuminating the development of ideas, and providing overviews to the literature concerning the covered topics. In most cases, these notes bring the reader to the research front. The text includes many figures that illustrate concepts and some parts of the proofs, enabling the reader to have a better understanding of the geometric meaning of the ideas. An appendix containing basic (and geometric) measure theory collects useful information for convex geometers.

Note: Preface -- 1. Convex functions -- 2. Convex sets -- 3. A first look into polytopes -- 4. Volume and area -- 5. Classical inequalities -- 6. Mixed volumes- 7. Mixed surface area measures -- 8. The Alexandrov-Frechel inequality -- 9. Affine convex geometry Part 1 -- 10. Affine convex geometry Part 2 -- 11. Further selected topics.-12. Historical steps of development of convexity as a field -- A. Measure theory for convex geometers -- References -- Index.

In: Springer Nature eBook

Additional Edition: Printed edition: ISBN 9783031505065

Additional Edition: Printed edition: ISBN 9783031505089

Language: English

DOI: 10.1007/978-3-031-50507-2

URL: https://doi.org/10.1007/978-3-031-50507-2

UID:

Format: 1 online resource (1195 pages)

Edition: 1st ed.

ISBN: 9783031505072

Series Statement: Cornerstones Series

Note: Intro -- Preface -- Contents -- 1 Convex functions -- 1.1 First steps -- 1.2 Regularity -- 1.3 Subgradients and subdifferentials -- 1.4 Duality -- 1.5 The Prékopa-Leindler inequality -- 1.6 Exercises -- 1.7 Notes -- 1.7.1 Properties of convex functions -- 1.7.2 The Prékopa-Leindler inequality and related topics -- 2 Convex sets -- 2.1 Basic notions -- 2.2 Support and separation -- 2.3 Gauge and support functions -- 2.4 Convex cones and boundary structure -- 2.5 The Hausdorff metric -- 2.6 Approximations and continuity -- 2.7 Steiner symmetrizations -- 2.8 Exercises -- 2.9 Notes -- 2.9.1 Special types of convex bodies -- 2.9.2 The Hausdorff metric in convexity -- 2.9.3 Steiner symmetrizations -- 3 A first look into polytopes -- 3.1 Basic concepts -- 3.2 Faces and normal cones -- 3.3 Strongly isomorphic polytopes -- 3.4 Approximations by polytopes -- 3.5 Exercises -- 3.6 Notes -- 3.6.1 Strongly isomorphic polytopes -- 3.6.2 Approximation of convex bodies by polytopes -- 4 Volume and area -- 4.1 Overview -- 4.2 Cavalieri's principle -- 4.3 Some formulas for volumes -- 4.4 Volumes and areas of polytopes -- 4.5 Measures on the boundary -- 4.6 Exercises -- 4.7 Notes -- 4.7.1 Cavalieri's principle -- 4.7.2 Volumes of unit balls and related topics -- 4.7.3 Volumes (of sections) of simplices -- 4.7.4 Volumes and areas of polytopes -- 5 Classical inequalities -- 5.1 The Brunn-Minkowski inequality -- 5.2 The isoperimetric inequality -- 5.3 Difference bodies and the Rogers-Shephard inequality -- 5.4 The Mahler product and the Blaschke-Santaló inequality -- 5.5 Mean width of a convex body and Urysohn's inequality -- 5.6 Exercises -- 5.7 Notes -- 5.7.1 The Brunn-Minkowski inequality -- 5.7.2 The isoperimetric inequality -- 5.7.3 Difference bodies and the Rogers-Shephard inequality -- 5.7.4 The Mahler product and the Blaschke-Santaló inequality. , 5.7.5 Mean width and Urysohn's inequality -- 6 Mixed volumes -- 6.1 Polynomiality of the volume -- 6.2 Properties of mixed volumes -- 6.3 Minkowski inequalities -- 6.4 Quermassintegrals and intrinsic volumes -- 6.5 Formulas for quermassintegrals -- 6.6 Exercises -- 6.7 Notes -- 6.7.1 Mixed volumes -- 6.7.2 Minkowski's inequalities related to mixed volumes -- 6.7.3 Quermassintegrals -- 7 Mixed surface area measures -- 7.1 Definition and properties -- 7.2 j-th order surface area measures -- 7.3 Minkowski's existence theorem -- 7.4 Surface area measures and curvature -- 7.5 Exercises -- 7.6 Notes -- 7.6.1 Minkowski's existence theorem and related topics -- 7.6.2 Surface area measures and curvature measures -- 8 The Alexandrov-Fenchel inequality -- 8.1 Wulff shapes -- 8.2 A proof of the inequality -- 8.3 Other versions and consequences -- 8.4 A brief discussion of the equality case -- 8.5 Exercises -- 8.6 Notes -- 8.6.1 Wulff shapes -- 8.6.2 The Alexandrov-Fenchel inequality -- 9 Affine convex geometry - Part 1 -- 9.1 John and Löwner ellipsoids -- 9.2 John's theorems for the centrally symmetric case -- 9.3 Minimal positions I: surface area and mean width -- 9.4 Minimal positions II: quermassintegrals -- 9.5 Exercises -- 9.6 Notes -- 9.6.1 Löwner-John ellipsoids and related topics -- 9.6.2 Positions -- 10 Affine convex geometry - Part 2 -- 10.1 Minkowski classes -- 10.2 Zonotopes and zonoids -- 10.3 Projection, centroid, and intersection bodies -- 10.4 Some related geometric inequalities -- 10.5 Exercises -- 10.6 Notes -- 10.6.1 Zonotopes -- 10.6.2 Zonoids -- 10.6.3 Projection, centroid, and intersection bodies -- 11 Further selected topics -- 11.1 Dual mixed volumes -- 11.2 The Lp Brunn-Minkowski theory -- 11.3 Valuations on convex bodies -- 11.4 Inequalities for covering numbers -- 11.5 The Bourgain-Milman inequality -- 11.6 Exercises -- 11.7 Notes. , 11.7.1 Dual mixed volumes -- 11.7.2 Valuations of convex bodies -- 11.7.3 The Bourgain-Milman inequality -- 12 Historical steps of development of convexity as a field -- 12.1 Greek antiquity -- 12.2 From the seventeenth to the nineteenth century -- 12.3 The twentieth century and the twenty-first century -- 12.3.1 Important contributors and authors of research monographs -- 12.3.2 Further books on convexity after 1950 -- 12.3.3 Problem books and problem-oriented surveys -- 12.3.4 Related proceedings -- 12.3.5 Handbooks -- 12.3.6 Selected and collected works -- 12.3.7 Examples of classical geometry books containing sections about convexity -- 12.3.8 A collection of surveys -- 12.4 Development of the overlaps with neighbouring disciplines -- 12.4.1 Discrete (and combinatorial) geometry of convex bodies -- 12.4.2 Polytopes, polyhedral sets, and related geometric figures -- 12.4.3 Convex algebraic geometry -- 12.4.4 Differential Geometry -- 12.4.5 Integral geometry and stochastic geometry -- 12.4.6 Convex analysis -- 12.4.7 Convexity and Banach spaces -- A Measure theory for convex geometers -- A.1 Outer measures and measures -- A.2 Measurable functions and integration -- A.3 Lebesgue measures -- A.4 Hausdorff measures -- A.5 Product measures -- A.6 Area and co-area formulas -- References -- Index.

Additional Edition: Print version: Balestro, Vitor Convexity from the Geometric Point of View Cham : Springer International Publishing AG,c2024 ISBN 9783031505065

Language: English

UID:

Format: 1 online resource (1195 pages)

Edition: 1st ed. 2024.

ISBN: 9783031505072

Series Statement: Cornerstones,

Content: This text gives a comprehensive introduction to the “common core” of convex geometry. Basic concepts and tools which are present in all branches of that field are presented with a highly didactic approach. Mainly directed to graduate and advanced undergraduates, the book is self-contained in such a way that it can be read by anyone who has standard undergraduate knowledge of analysis and of linear algebra. Additionally, it can be used as a single reference for a complete introduction to convex geometry, and the content coverage is sufficiently broad that the reader may gain a glimpse of the entire breadth of the field and various subfields. The book is suitable as a primary text for courses in convex geometry and also in discrete geometry (including polytopes). It is also appropriate for survey type courses in Banach space theory, convex analysis, differential geometry, and applications of measure theory. Solutions to all exercises are available to instructors who adopt the text for coursework. Most chapters use the same structure with the first part presenting theory and the next containing a healthy range of exercises. Some of the exercises may even be considered as short introductions to ideas which are not covered in the theory portion. Each chapter has a notes section offering a rich narrative to accompany the theory, illuminating the development of ideas, and providing overviews to the literature concerning the covered topics. In most cases, these notes bring the reader to the research front. The text includes many figures that illustrate concepts and some parts of the proofs, enabling the reader to have a better understanding of the geometric meaning of the ideas. An appendix containing basic (and geometric) measure theory collects useful information for convex geometers.

Note: Preface -- 1. Convex functions -- 2. Convex sets -- 3. A first look into polytopes -- 4. Volume and area -- 5. Classical inequalities -- 6. Mixed volumes- 7. Mixed surface area measures -- 8. The Alexandrov-Frechel inequality -- 9. Affine convex geometry Part 1 -- 10. Affine convex geometry Part 2 -- 11. Further selected topics.-12. Historical steps of development of convexity as a field -- A. Measure theory for convex geometers -- References -- Index.

Additional Edition: Print version: Balestro, Vitor Convexity from the Geometric Point of View Cham : Springer International Publishing AG,c2024 ISBN 9783031505065

Language: English

DOI: 10.1007/978-3-031-50507-2

UID:

Format: xix, 1184 Seiten

Edition: 1st ed. 2024

ISBN: 9783031505065

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-50506-5

Additional Edition: Erscheint auch als Druck-Ausgabe ISBN 978-3-031-50508-9

Additional Edition: Erscheint auch als Online-Ausgabe ISBN 978-3-031-50507-2

Language: English