UID:
almafu_9959240277202883
Umfang:
1 online resource (xiii, 490 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88436-0
,
0-511-94599-X
,
1-107-10261-8
,
1-107-09415-1
,
1-107-08794-5
,
0-511-52628-8
Serie:
Encyclopedia of mathematics and its applications ; volume 44
Inhalt:
At the heart of this monograph is the Brunn-Minkowski theory. It can be used to great effect in studying such ideas as volume and surface area and the generalizations of these. In particular the notions of mixed volume and mixed area arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered in detail. The author presents a comprehensive introduction to convex bodies and gives full proofs for some deeper theorems. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover; Half-title; Title; Copyright; Contents; Preface; 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; 1.7 The support function; 1.8 The Hausdorff metric; 2 Boundary structure; 2.1 Facial structure; 2.2 Singularities; 2.3 Segments in the boundary; 2.4 Polytopes; 2.5 Higher regularity and curvature; 2.6 Generic boundary structure; 3 Minkowski addition; 3.1 Minkowski addition and subtraction; 3.2 Summands and decomposition
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3.3 Approximation and addition3.4 Additive maps; 3.5 Zonoids and other classes of convex bodies; 4 Curvature measures and quermassintegrals; 4.1 Local parallel sets; 4.2 Curvature measures and area measures; 4.3 The area measure of order one; 4.4 Additive extension; 4.5 Integral-geometric formulae; 4.6 Local behaviour of curvature measures; 5 Mixed volumes and related concepts; 5.1 Mixed volumes and mixed area measures; 5.2 Extensions of mixed volumes; 5.3 Special formulae for mixed volumes and quermassintegrals; 5.4 Moment vectors and curvature centroids; 6 Inequalities for mixed volumes
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6.1 The Brunn-Minkowski theorem6.2 The Minkowski and isoperimetric inequalities; 6.3 The Aleksandrov-Fenchel inequality; 6.4 Consequences and improvements; 6.5 Generalized parallel bodies; 6.6 Equality cases and stability; 6.7 Linear inequalities; 6.8 Analogous notions and inequalities; 7 Selected applications; 7.1 Minkowski's existence theorem; 7.2 Uniqueness theorems for area measures; 7.3 The difference-body inequality; 7.4 Affinely associated bodies; Appendix Spherical harmonics; References; List of symbols; Author index; Subject index
,
English
Weitere Ausg.:
ISBN 0-521-05991-7
Weitere Ausg.:
ISBN 0-521-35220-7
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9780511526282
