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.
