Ebook: Selected Topics in Convex Geometry

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.

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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.

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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.

Content:
Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Metric Spaces....Pages 3-9
Subsets of Euclidean Space....Pages 11-24
Basic Properties of Convex Sets....Pages 25-37
Transformations of the Space K n of Compact Convex Sets....Pages 39-51
Rounding Theorems....Pages 53-59
Convex Polytopes....Pages 61-72
Functionals on the Space K n. The Steiner Theorem....Pages 73-87
The Hadwiger Theorems....Pages 89-95
Applications of the Hadwiger Theorems....Pages 97-105
Front Matter....Pages 107-107
Curvature and Surface Area Measures....Pages 109-123
Sets with positive reach. Convexity ring....Pages 125-134
Selectors for Convex Bodies....Pages 135-157
Polarity....Pages 159-171
Front Matter....Pages 173-173
Star Sets. Star Bodies....Pages 175-183
Intersection Bodies....Pages 185-191
Selectors for Star Bodies....Pages 193-201
Back Matter....Pages 203-226

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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.

Content:
Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Metric Spaces....Pages 3-9
Subsets of Euclidean Space....Pages 11-24
Basic Properties of Convex Sets....Pages 25-37
Transformations of the Space K n of Compact Convex Sets....Pages 39-51
Rounding Theorems....Pages 53-59
Convex Polytopes....Pages 61-72
Functionals on the Space K n. The Steiner Theorem....Pages 73-87
The Hadwiger Theorems....Pages 89-95
Applications of the Hadwiger Theorems....Pages 97-105
Front Matter....Pages 107-107
Curvature and Surface Area Measures....Pages 109-123
Sets with positive reach. Convexity ring....Pages 125-134
Selectors for Convex Bodies....Pages 135-157
Polarity....Pages 159-171
Front Matter....Pages 173-173
Star Sets. Star Bodies....Pages 175-183
Intersection Bodies....Pages 185-191
Selectors for Star Bodies....Pages 193-201
Back Matter....Pages 203-226
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