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El. knyga: Introduction to the Theory of Valuations

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Theory of valuations on convex sets is a classical part of convex geometry which goes back at least to the positive solution of the third Hilbert problem by M. Dehn in 1900. Since then the theory has undergone a multifaceted development. The author discusses some of Hadwiger's results on valuations on convex compact sets that are continuous in the Hausdorff metric. The book also discusses the Klain-Schneider theorem as well as the proof of McMullen's conjecture, which led subsequently to many further applications and advances in the theory. The last section gives an overview of more recent developments in the theory of translation-invariant continuous valuations, some of which turn out to be useful in integral geometry.

This book grew out of lectures that were given in August 2015 at Kent State University in the framework of the NSF CBMS conference ``Introduction to the Theory of Valuations on Convex Sets''. Only a basic background in general convexity is assumed.
Introduction 1(2)
Chapter 1 Basic definitions and examples
3(6)
1.1 Hadwiger's decomposition of a simplex
6(3)
Chapter 2 McMullen's decomposition theorem
9(6)
Chapter 3 Valuations on the line
15(6)
3.1 Hadwiger's characterization of n-homogeneous valuations
15(3)
3.2 Supporting functional and area measure
18(3)
Chapter 4 McMullen's description of (n -- 1)-homogeneous valuations
21(6)
4.1 Classification of valuations on the plane
25(2)
Chapter 5 The Klain--Schneider characterization of simple valuations
27(10)
5.1 Klain--Schneider characterization of simple valuations (even case)
27(5)
5.2 Klain--Schneider characterization of simple valuations (odd case)
32(5)
Chapter 6 Digression on the theory of generalized functions on manifolds
37(2)
Chapter 7 The Goodey--Weil imbedding
39(8)
Chapter 8 Digression on vector bundles
47(6)
8.1 Generalized sections of vector bundles and invariant form of the Goodey--Weil imbedding
51(2)
Chapter 9 The irreducibility theorem
53(8)
9.1 The Klain imbedding and the irreducibility theorem in the even case
53(1)
9.2 The Schneider imbedding and the irreducibility theorem in the odd case
54(7)
Chapter 10 Further developments
61(20)
10.1 Smooth translation-invariant valuations
61(1)
10.2 Normal cycle of convex sets and a presentation of smooth valuations
61(2)
10.3 Product on valuations
63(2)
10.4 Convolution of valuations
65(1)
10.5 Fourier type transform on valuations
66(1)
10.6 Pull-back and push-forward on valuations
67(2)
10.7 Valuations invariant under a group
69(2)
10.8 Valuations, Monge-Ampere operators, and non-commutative determinants
71(10)
Bibliography 81
Semyon Alesker, Tel Aviv University, Israel.
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