Integral Geometry and Algebraic Structures for Tensor Valuations

  • Andreas Bernig
  • Daniel Hug
Part of the Lecture Notes in Mathematics book series (LNM, volume 2177)


In this survey, we consider various integral geometric formulas for tensor-valued valuations that have been obtained by different methods. Furthermore we explain in an informal way recently introduced algebraic structures on the space of translation invariant, smooth tensor valuations, including convolution, product, Poincaré duality and Alesker-Fourier transform, and their relation to kinematic formulas for tensor valuations. In particular, we describe how the algebraic viewpoint leads to new intersectional kinematic formulas and substantially simplified Crofton formulas for translation invariant tensor valuations. We also highlight the connection to general integral geometric formulas for area measures.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of MathematicsGoethe-Universität FrankfurtFrankfurtGermany
  2. 2.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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