Selected Topics in Convex Geometry 1st Edition by Maria Moszyńska ISBN 978-0817643966 0817643966
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Product details:
ISBN 10: 0817643966
ISBN 13: 978-0817643966
Author: Maria Moszyńska
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. It 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:
Part I
1 Metric Spaces
1.1 Distance of point and set. Generalized balls.
1.2 The Hausdorff metric
2 Subsets of Euclidean Space
2.1 The Minkowski operations
2.2 Support hyperplane. The width
2.3 Convex sets
2.4 Compact convex sets. Convex bodies
2.5 Hyperplanes
3 Basic Properties of Convex Sets
3.1 Convex combinations
3.2 Convex hull
3.3 Metric projection
3.4 Support function
4 Transformations of the Space X” of Compact Convex Sets.
4.1 Isometries and similarities
4.2 Symmetrizations of convex sets. The Steiner symmetrization
4.3 Other symmetrizations
4.4 Means of rotations..
5 Rounding Theorems.
5.1 The first rounding theorem
5.2 Applications of the first rounding theorem
5.3 The second rounding theorem
5.4 Applications of the second rounding theorem
6 Convex Polytopes
6.1 Polyhedra and their role in topology
6.2 Convex polytopes
6.3 Approximation of convex bodies by polytopes
6.4 Equivalence by dissection
6.5 Spherical polytopes…
7 Functionals on the Space X”. The Steiner Theorem.
7.1 Functionals on the space K”
7.2 Basic functionals. The Steiner theorem
7.3 Consequences of the Steiner theorem
8 The Hadwiger Theorems
8.1 The first Hadwiger theorem
8.2 The second Hadwiger theorem
9 Applications of the Hadwiger Theorems
9.1 Mean width and mean curvature
9.2 The Crofton formulae
9.3 The Cauchy formulae
Part II
10 Curvature and Surface Area Measures
10.1 Curvature measures.
10.2 Surface area measures
10.3 Curvature and surface area measures for smooth, strictly convex bodies.
11 Sets with Positive Reach. Convexity Ring
11.1 Sets with positive reach
11.2 Convexity ring.
Contents
12 Selectors for Convex Bodies.
12.1 Symmetry centers
12.2 Selectors and multiselectors
12.3 Centers of gravity
12.4 The Steiner point..
12.5 Center of the minimal ring
12.6 Pseudocenter. G-pseudocenters.
12.7 G-quasi-centers. Chebyshev point
13 Polarity
13.1 Polar hyperplane of a point with respect to the unit sphere
13.2 Polarity for arbitrary subsets of R”
13.3 Polarity for convex bodies
13.4 Combinatorial duality induced by polarity
13.5 Santaló point
13.6 Self-duality of the center of the minimal ring
13.7 Metric polarity
Part III
14 Star Sets. Star Bodies
14.1 Star sets. Radial function
14.2 Star bodies
14.3 Radial metric
14.4 Star metric
15 Intersection Bodies
15.1 Dual intrinsic volumes
15.2 Projection bodies of convex bodies. The Shephard problem
15.3 Intersection bodies of star bodies. The Busemann-Petty problem.
15.4 Star duality
16 Selectors for Star Bodies
16.1 Radial centers of a star body
16.2 Radial centers of a convex body
16.3 Extended radial centers of a star body.
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Tags: Maria Moszyńska, Selected Topics, Convex Geometry