Convex valuation theory and integral geometry

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Convex geometry is a classical, yet rapidly developing branch of mathematics, appearing in various guises in a great variety of mathematical disciplines, ranging from algebraic and symplectic geometry, through probability theory and combinatorics, and on to algorithmics. Integral geometry appeared in the nineteenth century in several flavors. It distinguishes itself from differential geometry by studying global invariants (such as total length), rather than local invariants (such as curvature). My research concerns valuation theory. Valuation theory started with Dehn's solution of Hilbert's third problem on the definition of volume of polytopes in 3-dimensional space. In the simplest setting, valuations are continuously varying, finitely-additive measures on convex bodies. Many of the integral-geometric invariants of convex bodies, such as volume and surface area, are in fact valuations. Today, valuation theory is central in convex geometry, integral geometry, stochastic geometry and geometric probability. A natural problem in valuation theory is to find all valuations that are invariant under a group of symmetries. For example, volume and surface area remain unchanged during translations and rotations of a body. When the isotropy group of symmetries is compact, such as the group of rotations, a large toolbox, developed over the past 15 years, is readily available for the classification problem. I am interested primarily in non-compact groups of symmetries, where new approaches must be developed. The famous isoperimetric inequality has a far reaching generalization, the Alexandrov-Fenchel inequality, which concerns valuations of a particular kind known as mixed volumes. It is therefore natural to look for other geometric inequalities involving valuations. Another central player in integral geometry is the Radon transform, which nominally replaces a function by its integrals over lines, and forms the basis of the field of tomography, used in medical imaging, seismography etc. Valuation theory offers a vast generalization of the Radon transform, which I plan to explore.