Convex Bodies: The Brunn-Minkowski TheoryAt 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 that have never previously been brought together. Many hints and pointers to connections with other fields are given, and an exhaustive reference list is included. |
Contents
1 Basic convexity | 1 |
2 Boundary structure | 62 |
3 Minkowski addition | 126 |
4 Curvature measures and quermassintegrals | 197 |
5 Mixed volumes and related concepts | 270 |
6 Inequalities for mixed volumes | 309 |
7 Selected applications | 389 |
Appendix Spherical harmonics | 428 |
References | 433 |
List of symbols | 474 |
Author index | 478 |
| 484 | |
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Common terms and phrases
affine Aleksandrov arbitrary assertion assume ball Blaschke Borel set boundary point Brunn-Minkowski theorem centrally symmetric centroid class C² closed convex compact contains conv converges convex bodies convex function convex hull convex sets curvature measures deduce defined denote differentiable equality holds exists extended Fenchel finite formula geometry given Groemer Hadwiger halfspace Hausdorff Hausdorff measure hence homothetic hyperplane implies indecomposable integral intersection isoperimetric inequality K₁ Lebesgue measure Lemma linear subspace Lutwak metric Minkowski addition Minkowski sums mixed area measure mixed volumes n-dimensional nonempty normal vector Notes for Section obtain orthogonal polytopes proof of Theorem properties proved quermassintegrals relint result rigid motion satisfies Schneider SO(n space Steiner point strongly isomorphic subset summand support function support plane translate unique unit normal vector zonoids zonotope