**CBMS Regional Conference Series in Mathematics**

Volume: 126;
2018;
83 pp;
Softcover

MSC: Primary 52;

Print ISBN: 978-1-4704-4359-7

Product Code: CBMS/126

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

**Electronic ISBN: 978-1-4704-4717-5
Product Code: CBMS/126.E**

List Price: $52.00

AMS Member Price: $41.60

MAA Member Price: $46.80

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#### Supplemental Materials

# Introduction to the Theory of Valuations

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*Semyon Alesker*

A co-publication of the AMS and CBMS

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.

#### Readership

Graduate students and researchers interested in the theory of valuations on convex sets.

#### Reviews & Endorsements

For the newcomer to the field, the book gives a first orientation, insights into the underlying structures, and valuable hints to the original literature.

-- Rolf Schneider, Zentralblatt MATH

This little book gives a very nice overview of a subject nearly at the cutting edge of research.

-- P. McMullen, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to the Theory of Valuations

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction 18
- Chapter 1. Basic definitions and examples 310
- Chapter 2. McMullen’s decomposition theorem 916
- Chapter 3. Valuations on the line 1522
- Chapter 4. McMullen’s description of (𝑛-1)-homogeneous valuations 2128
- Chapter 5. The Klain–Schneider characterization of simple valuations 2734
- Chapter 6. Digression on the theory of generalized functions on manifolds 3744
- Chapter 7. The Goodey–Weil imbedding 3946
- Chapter 8. Digression on vector bundles 4754
- Chapter 9. The irreducibility theorem 5360
- Chapter 10. Further developments 6168
- 10.1. Smooth translation-invariant valuations 6168
- 10.2. Normal cycle of convex sets and a presentation of smooth valuations 6168
- 10.3. Product on valuations 6370
- 10.4. Convolution of valuations 6572
- 10.5. Fourier type transform on valuations 6673
- 10.6. Pull-back and push-forward on valuations 6774
- 10.7. Valuations invariant under a group 6976
- 10.8. Valuations, Monge-Ampère operators, and non-commutative determinants 7178

- Bibliography 8188
- Back Cover Back Cover193