Geometry and Algebraic Structures

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, Poincare 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.