| Description |
1 online resource (760 pages) |
| Series |
Encyclopedia of Mathematics and its Applications ; v. 151 |
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Encyclopedia of mathematics and its applications.
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| Contents |
Preface to the second edition; Preface to the first edition; General hints to the literature; Conventions and notation; 1 Basic convexity; 1.1 Convex sets and combinations; 1.2 The metric projection; 1.3 Support and separation; 1.4 Extremal representations; 1.5 Convex functions; 1.6 Duality; 1.6.1 Duality of convex sets; 1.6.2 Duality of convex functions; 1.7 Functions representing convex sets; 1.7.1 The support function; 1.7.2 Further representing functions; 1.8 The Hausdorff metric; 2 Boundary structure; 2.1 Facial structure; 2.2 Singularities; 2.3 Segments in the boundary |
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2.4 Polytopes2.5 Higher regularity and curvature; 2.6 Generalized curvatures; 2.7 Generic boundary structure; 3 Minkowski addition; 3.1 Minkowski addition and subtraction; 3.2 Summands and decomposition; 3.3 Additive maps; 3.4 Approximation and addition; 3.5 Minkowski classes and additive generation; 4 Support measures and intrinsic volumes; 4.1 Local parallel sets; 4.2 Steiner formula and support measures; 4.3 Extensions of support measures; 4.4 Integral-geometric formulae; 4.5 Local behaviour of curvature and area measures; 5 Mixed volumes and related concepts |
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5.1 Mixed volumes and mixed area measures5.2 Extensions of mixed volumes; 5.3 Special formulae for mixed volumes; 5.3.1 Formulae involving curvature functions; 5.3.2 Formulae involving integrations; 5.3.3 Formulae for generalized zonoids; 5.4 Moment vectors, curvature centroids, Minkowski tensors; 5.4.1 Moment vectors and curvature centroids; 5.4.2 Minkowski tensors; 5.5 Mixed discriminants; 6 Valuations on convex bodies; 6.1 Basic facts and examples; 6.2 Extensions; 6.3 Polynomiality; 6.4 Translation invariant, continuous valuations; 6.5 The modern theory of valuations |
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7 Inequalities for mixed volumes7.1 The Brunn-Minkowski theorem; 7.2 The Minkowski and isoperimetric inequalities; 7.3 The Aleksandrov-Fenchel inequality; 7.4 Consequences and improvements; 7.5 Wulff shapes; 7.6 Equality cases and stability; 7.7 Linear inequalities; 8 Determination by area measures and curvatures; 8.1 Uniqueness results; 8.2 Convex bodies with given surface area measures; 8.2.1 Minkowski's existence theorem; 8.2.2 Blaschke addition; 8.3 The area measure of order one; 8.3.1 The length measure in the plane; 8.3.2 The Christoffel problem; 8.4 The intermediate area measures |
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8.5 Stability and further uniqueness results9 Extensions and analogues of the Brunn-Minkowski theory; 9.1 The Lp Brunn-Minkowski theory; 9.2 The Lp Minkowski problem and generalizations; 9.3 The dual Brunn-Minkowski theory; 9.4 Further combinations and functionals; 9.5 Log-concave functions and generalizations; 9.6 A glimpse of other ramifications; 10 Affine constructions and inequalities; 10.1 Covariogram and difference body; 10.2 Qualitative characterizations of ellipsoids; 10.3 Steiner symmetrization; 10.4 Shadow systems; 10.5 Curvature images and affine surface areas |
| Summary |
A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research |
| Notes |
Print version record |
| Subject |
Convex bodies.
|
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Convex bodies.
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| Form |
Electronic book
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| ISBN |
9781107465527 |
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1107465524 |
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