Introduction to the Theory of Valuations
About this Title
Semyon Alesker, Tel Aviv University, Tel Aviv, Israel
Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 2018; Volume 126
ISBNs: 978-1-4704-4359-7 (print); 978-1-4704-4717-5 (online)
MathSciNet review: MR3820854
MSC: Primary 52B45; Secondary 52A39
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.
Graduate students and researchers interested in the theory of valuations on convex sets.
Table of Contents
- Basic definitions and examples
- McMullen’s decomposition theorem
- Valuations on the line
- McMullen’s description of $(n-1)$-homogeneous valuations
- The Klain-Schneider characterization of simple valuations
- Digression on the theory of generalized functions on manifolds
- The Goodey-Weil imbedding
- Digression on vector bundles
- The irreducibility theorem
- Further developments