Introduction to the theory of valuations
Author(s)
Bibliographic Information
Introduction to the theory of valuations
(Regional conference series in mathematics, no. 126)
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, c2018
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
"Support from the National Science Foundation."
NSF-CBMS Regional Conference in the Mathematical Sciences on Introduction to the Theory of Valuations on Convex Sets, held at Kent State University, August 10-15, 2015
Includes bibliographical references
Description and Table of Contents
Description
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.
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
Bibliography
by "Nielsen BookData"