Affine isoperimetric inequalities

The classical Euclidean isoperimetric inequality is widely considered the most fundamental inequality of geometric analysis, with many applications in both pure and applied mathematics. In common terms it explains why the surface area of a soap bubble, for which the volume of air inside does not change, is minimized when the bubble is spherical. The mathematical statement of the inequality holds in much more generality, for example also in two or more than three dimensions. Over the last two decades a number of isoperimetric type inequalities for geometric quantities, which are invariant not only under translations and rotations but also larger groups of transformations, have been established. Such inequalities have become known as affine isoperimetric inequalities. The exciting recent developments in the theory of such inequalities made it also more and more evident that inequalities for affine invariant quantities are more powerful than their often better known Euclidean relatives. These recent advances were in many cases intimately tied to the Lp extension of the classical Brunn-Minkowski theory of convex sets. An aim of the proposed research program is, to use new tools that have just become available, to attack some of the open problems that arose in the Lp Brunn-Minkowski theory, in particular, for the critical values p < 1. This includes work on the long sought after Lp extension of Busemann`s intersection inequality as well as the recently conjectured log-Brunn-Minkowski inequality. For a long time the theory of affine isoperimetric inequalities was restricted by the shackles of flat space. Recently, however, a version of the famous Blaschke-Santal inequality was obtained on the Euclidean unit sphere and analogues of Blaschke`s affine surface area have been discovered in spherical and hyperbolic space. A main goal of this proposed program is to systematically generalize affine invariants to non-Euclidean spaces, establish isoperimetric inequalities for these quantities and relate the new inequalities to classical ones. In fact, there has never been a better time to invest in this line of research, where any breakthrough has the potential not only to reshape our understanding of classical affine isoperimetric inequalities, but also to open the door to a wide array of new applications to PDEs, Banach space geometry, and geometric tomography.

The classical Euclidean isoperimetric inequality, explaining for example why the surface area of a soap bubble, for which the volume of air inside does not change, is minimized when the bubble is spherical, is widely considered the most fundamental inequality of geometric analysis. Over the last decades a number of isoperimetric type inequalities for geometric quantities, which are invariant not only under translations and rotations but also larger groups of affine or linear transformations, have been established. Such inequalities have become known as affine isoperimetric inequalities. The main results obtained in the project can be roughly divided into two related categories. On the one it was shown that many classical inequalities from Euclidean and affine geometry hold for much more general classes of geometric quantities than was previously understood. These new isoperimetric inequalities either directly provide fundamental relations between certain (invariant) valuations, such as the Hodge-Riemann relations, or the geometric functionals involved in the inequalities are derived from so called Minkowski valuations. On the other hand the restriction of some affine isoperimetric inequalities to the shackles of flat space could be lifted. More precisely, a spherical analogue of the polar Busemann-Petty centroid inequality has been discovered and a randomized version of the spherical Blaschke-Santal inequality was also obtained. Additionally, the full strength of affine inequalities compared to their Euclidean counterparts was revealed in several instances. For example, the celebrated Blaschke-Santal inequality, Petty`s projection inequality, and the recently established inequalities for affine quermassintegrals were shown to be significantly stronger than large families of isoperimetric inequalities for Minkowski valuations which intertwine rigid motions. The main idea in the proofs of these results was to exploit new convolution representations for Minkowski valuations that turned out to be tailor made for the application of tools from harmonic and functional analysis