Minkowski valuations and geometric inequalities

As a generalization of the notion of measure, valuations on convex sets have always played a central role in geometry. A particularly exciting new development in the theory of valuations explores the strong connections between convex body valued valuations and the theory of affine isoperimetric and analytic inequalities. To be more specific, many powerful affine isoperimetric inequalities involve linearly intertwining Minkowski valuations. Although a large part of the theory of convex body valued valuations deals with operators compatible with linear transformations, considerable effort has been invested in recent years to also classify all continuous rigid motion compatible Minkowski valuations. These results in turn have applications to Brunn-Minkowski type inequalities which were shown to hold for various large classes of valuations equivariant under orthogonal transformations. A goal of this project is to further clarify which of the classical (and more recent) affine geometric inequalities can be generalized to functionals derived from Minkowski valuations which are compatible with orthogonal transformations only. Analytic descriptions of these Minkowski valuations will play a key role in these efforts. The theory of valuations is deeply intertwined with the Brunn-Minkowski theory which arises from combining the notion of Minkowski addition of convex bodies with that of ordinary volume. Two decades ago, a combination of an old notion of Minkowski-Firey Lp addition with volume led to an embryonic Lp Brunn-Minkowski theory. Over the last 20 years this rapidly growing theory has built strong ties with the theory of valuations. For example an Lp analog of the classical projection operator was introduced and an important Lp extension of one of the fundamental affine isoperimetric inequalities, the Petty projection inequality, was established. This extension is the core of a sharp affine Lp Sobolev inequality which strengthens the classical Lp Sobolev inequality. However, the advances in valuation theory revealed that the Lp projection operator is only one representative of an entire class of Lp extensions of the classical projection operator. This fundamental result has very recently led to a further generalization of Petty`s projection inequality and a new asymmetric affine Lp Sobolev inequality. The well known equivalence of the isoperimetric inequality and the sharp Sobolev inequality is an important example of the interplay between analytic and geometric inequalities. This remarkable link has been amplified by the recent work on affine analytic inequalities. It is one of the aims of this project to further exploit these strong relations and to establish new affine log-Sobolev and Gagliardo-Nirenberg inequalities.

The famous BrunnMinkowski inequality expresses the fundamental fact that the volume functional is log-concave and directly implies the classical Euclidean isoperimetric inequality. Only in the second half of the last century was it discovered that an affine projection inequality of Petty, involving Minkowskis projection bodies, is not just significantly stronger than the isoperimetric inequality, but in fact an optimal version of it. The underlying reason for the special role of Pettys projection inequality has been demonstrated recently, when the projection body map was characterized as the unique convex body valued valuation which is contravariant with respect to linear transformations.As a generalization of measures, real valued valuations have long been at the center of convex geometric analysis. Minkowski valuations, which are valuations taking values in the space of convex bodies endowed with Minkowski addition, are of newer vintage and build a bridge between the theory of (affine) isoperimetric inequalities and modern integral geometry. A central theme of this research project was to gain a deeper understanding of Minkowski valuations compatible with volume preserving linear maps on the one hand and Minkowski valuations intertwining rigid motions on the other hand. In both cases new classification results not only enhanced our understanding of the fundamental characteristics of classical operators but also led to the discovery of new convex body valued valuations. In the case of Minkowski valuations compatible with rigid motions these results have laid the groundwork for a new structure theory of these valuations to be developed further in upcoming years.The new characterization theorems for Minkowski valuations, in turn, played a key role in the discovery of new geometric inequalities and the strengthening of classical inequalities. By systematically exploiting new integral representations of Minkowski valuations, it became clear that various classical inequalities hold in a more general setting (that is, for much larger classes of operators). Also the full strength of affine inequalities (from geometry and analysis) compared to their Euclidean counterparts was further illuminated. For example, well known log-concavity properties of the intrinsic volumes and the intrinsic volumes of projection bodies were shown to hold for all Minkowski valuations intertwining rigid motions and a recent asymmetric Lp version of Pettys projection inequality was used to establish a new affine Plya-Szegö principle which allows for effortless proofs of affine Sobolev and log-Sobolev inequalities. Moreover, reverse isoperimetric inequalities for Wulff shapes were obtained which generalize the celebrated Ball-Barthe volume ratio inequalities.