Linearly intertwining maps on convex bodies

Functions defined on convex bodies which are compatible with the general linear group lie at the very core of convex geometric analysis, Minkowski geometry, stochastic geometry, and geometric tomography. Of fundamental importance for the study of such functions is the concept of valuations. Valuations are generalizations of measures and played a key role in Dehn`s solution of Hilbert`s Third Problem. Since then valuations have always been a central part of geometry. Over the last years, important functions from the classical Brunn-Minkowski theory were characterized as valuations which are compatible with the whole general linear group. Examples include the affine surface area as well as the projection-, centroid-, and intersection body operator. Recently, some of these results were generalized. It turned out that valuations which are linearly intertwining only with respect to the special linear group can also be completely classified. Moreover, these classifications revealed a whole variety of new notions which are much more general than previously known ones. Therefore, a thorough study of valuations which are linearly intertwining only with respect to proper subgroups of the general linear group is proposed. The development of Lp extensions of the Brunn-Minkowski theory constitutes a major part of modern convex geometric analysis. This Lp Brunn-Minkowski theory witnessed an explosive growth over the last years. Inequalities of this Lp theory turn out to almost invariably be stronger than their classical counterparts. The above mentioned characterizations have been used to determine the correct Lp analogs of projection-, centroid-, and intersection bodies. Surprisingly, in each case there is not just one but a whole family of such Lp analogs. This insight led to new affine isoperimetric inequalities induced by linearly intertwining operators. Moreover, these geometric inequalities were used to establish a new affine Lp Sobolev inequality which strengthens and directly implies the classical sharp Lp Sobolev inequality. Further research on geometric inequalities induced by linearly intertwining operators is proposed. The deepening of connections between such inequalities and analysis is also part of this project. As was mentioned before, recently discovered linearly intertwining functions led to both extensions and strengthenings of classical inequalities. This demonstrates the clear need to move to the next evolutionary development of the Brunn-Minkowski theory: An Orlicz Brunn-Minkowski theory. Very recently, some elements of this Orlicz Brunn-Minkowski theory have been uncovered, namely Orlicz projection- and centroid bodies as well as an Orlicz Minkowski problem. However, much work remains to be done. The task of finding additional elements of the Orlicz Brunn-Minkowski theory is part of the proposed work.

The concept of valuation lies at the heart of geometry. A valuation is a function defined on convex sets that is additive with respect to unions and intersections. Volume is an important example. Among the numerous further examples are the surface area and more generally the intrinsic volumes as well as the affine surface area, projection bodies, and intersection bodies. Valuations arise naturally in many problems. Applications in integral geometry and geometric probability are classical. More recently, the connection to problems in analysis and the theory of Sobolev inequalities has been established. For these applications, results on the classification of valuations are most useful. This is the central subject of the project. In particular, such classifications have been established for valuations that intertwine the special linear group, SL(n). Important examples are again the volume but also so called Orlicz affine surface areas and Minkowski valuations that associate with a convex set another convex set. Among the main results established in this project is a complete classification of continuous and SL(n) invariant valuations on convex sets containing the origin in their interiors and a complete classification of measure valued valuations and a characterization of Lp curvature measures.