Isoperimetric Inequalities and Integral Geometry

In recent years the theory of affine isoperimetric inequalities has become deeply intertwined with the theory of valuations, which is an important part of modern integral geometry. The bridge between these previously unrelated areas is built on the fact that many powerful affine inequalities involve basic operators on convex bodies which intertwine linear transformations, e.g., projection and intersection body maps. The underlying reason for the special role of these operators has been demonstrated only recently, when they were given characterizations as the unique valuations which are compatible with affine transformations. Moreover, through these characterization results new convex and star body valued valuations were discovered which subsequently led to a strengthening of a number of affine isoperimetric inequalities. These inequalities in turn form the geometric core of new sharp affine analyic inequalities, e.g., affine Sobolev and log-Sobolev inequalities. Although a large part of the theory of convex body valued valuations deals with operators intertwining volume preserving linear maps, considerable effort has also been invested to classify all continuous and rigid motion compatible body valued valuations. These results in turn shed a new light on various affine geometric inequalities arising from linearly intertwining valuations, since they were shown to hold for much larger classes of valuations intertwining rigid motions only. The results obtained here so far appear to be only the tip of an iceberg. An aim of this project is to systematically exploit the valuation point of view to uncover the bigger picture beneath and reshape our understanding of many fundamental affine isoperimetric inequalities. Not only should it become clear that these inequalities hold in a more general setting (that is, for much larger classes of operators) but also the full strength of affine inequalities (in geometry and analysis) compared to their Euclidean counterparts should be illuminated. New characterization theorems for convex body valued valuations will play a key role in these efforts. The theory of translation invariant scalar valued valuations has seen a series of striking developments over the last years. In particular, through the introduction of new algebraic structures substantial inroads have been made towards a fuller understanding of the integral geometry of groups acting transitively on the sphere. A goal of this project is to introduce a corresponding algebraic machinery for convex body valued valuations which would provide the means to attack some of the major open problems in the area of affine isoperimetric inequalities. Here the new idea is to consider these questions in the right, larger setting of valuations intertwining rigid motions and exploit the fact that more algebraic structure is present in this class of operators.

The core objective of this START project was to gain a deeper understanding of the connections between the theory of isoperimetric inequalities and valuation theory, which is central to modern integral geometry and concerns the study of geometric functionals that allow computation by parts. The bridge between these previously unrelated areas is built on the fact that many powerful isoperimetric inequalities in convex geometric analysis either directly provide fundamental relations between certain (invariant) valuations or the geometric functionals involved in the inequalities are derived from valuations. In the latter case, the valuations in question are mainly convex body valued operators defined by tomographic data, such as projections and sections, which intertwine volume preserving affine maps or, merely, rigid motions. The main results obtained in the project can be roughly divided into two interrelated categories. On the one hand, new classification and representation theorems for convex body valued valuations illuminated large classes of natural geometric objects which give rise to new invariants. Thereby, several well studied quantities were generalized and the significance of classical invariants could be underlined. On the other hand, a systematic application of the new integral representations for valuations shed a new light on various affine geometric inequalities arising from linearly intertwining valuations, since they were shown to hold for much larger classes of valuations intertwining rigid motions only. Moreover, the full strength of affine inequalities (from geometry and analysis) compared to their Euclidean counterparts was revealed. For example, the celebrated BlaschkeSantalo inequality and Pettys projection inequality, the latter being significantly stronger than the classical isoperimetric inequality, were shown to hold for a large family of socalled Minkowski valuations which intertwine rigid motions only. In turn, these new isoperimetric inequalities allowed for effortless proofs of Sobolev type inequalities that strengthen the classical sharp Sobolev inequality. One major new idea in the proofs of several results was to exploit the rich algebraic structure on translation invariant scalar valuations in the context of convex body valued valuations compatible with rigid motions. In particular, the Alesker product and generalized valuations were critical for establishing a new convolution representation for Minkowski valuations, while a derivation operator on such valuations was used in the solution of a variety of isoperimetric problems that were made accessible only by considering them in the right, sufficiently large context.