Selected Topics in Convex Geometry

by Moszynska, Maria

• ISBN13: 9780817643966
• ISBN10: 0817643966
• Format: Paperback
• Copyright: 2005-12-01
• Publisher: Birkhauser

Summary

The field of convex geometry has become a fertile subject of mathematical activity in the past few decades. This exposition, examining in detail those topics in convex geometry that are concerned with Euclidean space, is enriched by numerous examples, illustrations, and exercises, with a good bibliography and index. The theory of intrinsic volumes for convex bodies, along with the Hadwiger characterization theorems, whose proofs are based on beautiful geometric ideas such as the rounding theorems and the Steiner formula, are treated in Part 1. In Part 2 the reader is given a survey on curvature and surface area measures and extensions of the class of convex bodies. Part 3 is devoted to the important class of star bodies and selectors for convex and star bodies, including a presentation of two famous problems of geometric tomography: the Shephard problem and the Busemann-Petty problem. Selected Topics in Convex Geometry requires of the reader only a basic knowledge of geometry, linear algebra, analysis, topology, and measure theory. The book can be used in the classroom setting for graduates courses or seminars in convex geometry, geometric and convex combinatorics, and convex analysis and optimization. Researchers in pure and applied areas will also benefit from the book.

Table of Contents

Preface to the Polish Edition	ix
Preface to the English Edition	xi
Introduction	xiii
Part I
1 Metric Spaces	3	(8)
1.1 Distance of point and set. Generalized balls	3	(3)
1.2 The Hausdorff metric	6	(5)
2 Subsets of Euclidean Space	11	(14)
2.1 The Minkowski operations	11	(3)
2.2 Support hyperplane. The width	14	(4)
2.3 Convex sets	18	(1)
2.4 Compact convex sets. Convex bodies	19	(1)
2.5 Hyperplanes	20	(5)
3 Basic Properties of Convex Sets	25	(14)
3.1 Convex combinations	25	(3)
3.2 Convex hull	28	(3)
3.3 Metric projection	31	(3)
3.4 Support function	34	(5)
4 Transformations of the Space Kn of Compact Convex Sets	39	(14)
4.1 Isometries and similarities	39	(2)
4.2 Symmetrizations of convex sets. The Steiner symmetrization	41	(7)
4.3 Other symmetrizations	48	(1)
4.4 Means of rotations	49	(4)
5 Rounding Theorems	53	(8)
5.1 The first rounding theorem	53	(2)
5.2 Applications of the first rounding theorem	55	(1)
5.3 The second rounding theorem	56	(2)
5.4 Applications of the second rounding theorem	58	(3)
6 Convex Polytopes	61	(12)
6.1 Polyhedra and their role in topology	61	(3)
6.2 Convex polytopes	64	(3)
6.3 Approximation of convex bodies by polytopes	67	(2)
6.4 Equivalence by dissection	69	(2)
6.5 Spherical polytopes	71	(2)
7 Functionals on the Space Kn. The Steiner Theorem	73	(16)
7.1 Functionals on the space Kn	73	(4)
7.2 Basic functionals. The Steiner theorem	77	(5)
7.3 Consequences of the Steiner theorem	82	(7)
8 The Hadwiger Theorems	89	(8)
8.1 The first Hadwiger theorem	89	(6)
8.2 The second Hadwiger theorem	95	(2)
9 Applications of the Hadwiger Theorems	97
9.1 Mean width and mean curvature	97	(1)
9.2 The Crofton formulae	98	(4)
9.3 The Cauchy formulae	102	(7)
Part II
10 Curvature and Surface Area Measures	109	(16)
10.1 Curvature measures	109	(8)
10.2 Surface area measures	117	(3)
10.3 Curvature and surface area measures for smooth, strictly convex bodies	120	(5)
11 Sets with Positive Reach. Convexity Ring	125	(10)
11.1 Sets with positive reach	125	(2)
11.2 Convexity ring	127	(8)
12 Selectors for Convex Bodies	135	(24)
12.1 Symmetry centers	135	(2)
12.2 Selectors and multiselectors	137	(1)
12.3 Centers of gravity	138	(4)
12.4 The Steiner point	142	(2)
12.5 Center of the minimal ring	144	(5)
12.6 Pseudocenter. G-pseudocenters	149	(5)
12.7 G-quasi-centers. Chebyshev point	154	(5)
13 Polarity	159	(16)
13.1 Polar hyperplane of a point with respect to the unit sphere	159	(2)
13.2 Polarity for arbitrary subsets of Rn	161	(2)
13.3 Polarity for convex bodies	163	(2)
13.4 Combinatorial duality induced by polarity	165	(1)
13.5 SantalO point	166	(2)
13.6 Self-duality of the center of the minimal ring	168	(1)
13.7 Metric polarity	169	(6)
Part III
14 Star Sets. Star Bodies	175	(10)
14.1 Star sets. Radial function	175	(2)
14.2 Star bodies	177	(2)
14.3 Radial metric	179	(2)
14.4 Star metric	181	(4)
15 Intersection Bodies	185	(1)
15.1 Dual intrinsic volumes	185	(8)
15.2 Projection bodies of convex bodies. The Shephard problem	186	(1)
15.3 Intersection bodies of star bodies. The Busemann–Petty problem	187	(2)
15.4 Star duality	189	(4)
16 Selectors for Star Bodies	193	(10)
16.1 Radial centers of a star body	193	(2)
16.2 Radial centers of a convex body	195	(3)
16.3 Extended radial centers of a star body	198	(5)
Exercises to Part I	203	(8)
Exercises to Part II	211	(4)
Exercises to Part III	215	(2)
References	217