Among several trends in convex geometric analysis, two have undergone an explosive development in recent years: the theory of affine isoperimetric and analytic inequalities, and the enhanced understanding of fundamental concepts of the subject as a whole lent by the theory of valuations. The project concerns both of these trends.
The connections between convex body valued valuations and isoperimetric inequalities (like, the Petty projection inequality or affine Sobolev inequalities and their extensions) have attracted the interest of first-rate research groups in the world. However, the underlying bigger picture behind these strong relations has yet to be discovered. A goal of the proposed research program is to systematically exploit the “valuations point of view” to reshape not only the way (affine) isoperimetric inequalities are thought of and applied but also the way these powerful inequalities are established.
Through the introduction of new algebraic structures on the space of translation invariant scalar valuations substantial inroads have been made towards a fuller understanding of the integral geometry of groups acting transitively on the sphere. An aim of the project is to introduce a corresponding algebraic machinery in the theory of convex body valued valuations which would provide the means to attack long standing major open problems in the area of affine isoperimetric inequalities.
It is our strong belief that over the next years it will become clear that many classical inequalities from affine geometry hold in a much more general setting than is currently understood. This will not only lead to the discovery of new inequalities but also should reveal the full strength of affine inequalities compared to their counterparts from Euclidean geometry. The proposed research goals of this ERC Grant Project would therefore represent a huge step towards advancing these developments that will alter two main subjects at the same time.