Integral Geometry and Valuations

1. Front Matter

   Pages i-viii

   PDF

2. New Structures on Valuations and Applications

   • Semyon Alesker

   Pages 1-45

3. Algebraic Integral Geometry

   • Joseph H. G. Fu

   Pages 47-112

4. Back Matter

   Pages 113-113

   PDF

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, provides an introduction to the theory of convex valuations with emphasis on recent developments. In particular, it presents the new structures on the space of valuations discovered after Alesker's irreducibility theorem. The newly developed theory of valuations on manifolds is also described. In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santaló. The approach is new and based on the notions and tools presented in the first part. This original viewpoint not only enlightens the classical integral geometry of euclidean space, but it also allows the computation of kinematic formulas in other geometries, such as hermitian spaces. The book will appeal to graduate students and interested researchers from related fields including convex, stochastic, and differential geometry. ​

• Hermitian spaces
• convex sets
• irreducibility theorem
• kinematic formulas
• valuations

• Universitat Autònoma de Barcelona, Barcelona, Spain

  Eduardo Gallego, Gil Solanes

• Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

  Semyon Alesker

• Department of Mathematics, University of Georgia, Athens, USA

  Joseph H.G. Fu

Policies and ethics