Integral Geometry and Valuations

  • Semyon Alesker
  • Joseph H.G. Fu
  • Eduardo Gallego
  • Gil Solanes

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Joseph H. G. Fu
    Pages 47-112
  3. Back Matter
    Pages 113-113

About this book


Valuations are finitely additive functionals on the space of convex bodies. Their study has become a central subject in convexity theory, with fundamental applications to integral geometry. In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This book originated from two courses delivered by the authors at the CRM and provides a self-contained introduction to these topics, covering most of the recent advances.

The first part, by Semyon Alesker, is devoted to the theory of convex valuations, with emphasis on the latest developments. A special focus is put on the new fundamental structures of the space of valuations discovered after Alesker's irreducibility theorem. Moreover, the author describes the newly developed theory of valuations on manifolds.

In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santaló, based on the notions and tools presented in the first part. At the core of this approach lies the close relationship between kinematic formulas and Alesker's product of valuations. This original viewpoint not only enlightens the classical integral geometry of Euclidean space, it has also produced previously unreachable results, such as the explicit computation of kinematic formulas in Hermitian spaces.


Hermitian spaces convex sets irreducibility theorem kinematic formulas valuations

Authors and affiliations

  • Semyon Alesker
    • 1
  • Joseph H.G. Fu
    • 2
  1. 1.Department of MathematicsTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

Editors and affiliations

  • Eduardo Gallego
    • 1
  • Gil Solanes
    • 2
  1. 1.Universitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Universitat Autònoma de BarcelonaBarcelonaSpain

Bibliographic information

  • DOI
  • Copyright Information Springer Basel 2014
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0873-6
  • Online ISBN 978-3-0348-0874-3
  • Series Print ISSN 2297-0304
  • Series Online ISSN 2297-0312
  • Buy this book on publisher's site