Hessian inequalities and extensions to Sobolev spaces

Among all geometric figures in the plane, the circle has the property that it has the largest possible area for a given circumference. Similarly, in three-dimensional space, a sphere has the largest volume for a given surface area. In mathematics, this fact is generally expressed by the so-called isoperimetric inequality, which is a special case of even more general volume inequalities. Recently, the concept of surface area and similar related characteristics has been extended from geometric figures or bodies to convex functions. This is known as Hessian valuations. The aim of the project is to generalize the volume inequalities which are known for geometric objects to the new Hessian valuations. Furthermore, both the Hessian valuations as well as the new inequalities are to be extended to an even larger class of functions. One anticipates that this will not only lead to new insights and applications for functions themselves, but also to new approaches to problems related to the classical volume inequalities.

Volume and surface area are natural quantities that one can assign to a given shape to measure its size. These quantities are generalized by the intrinsic volumes which are central objects in convex and integral geometry, where so-called convex bodies are studied. Recently, functional intrinsic volumes on convex functions were introduced. These operators associate a number to a given convex function and serve as functional analogs of the intrinsic volumes. The premise of this project is based on the fact that the classical intrinsic volumes satisfy some well-known significant inequalities, such as the isoperimetric inequality. It is therefore only natural to ask whether similar inequalities also hold for their new functional counterparts. While it turned out that such conjectured inequalities do not hold in the functional world, the outcomes of this project still provide many new exciting insights into functional intrinsic volumes. One major result of this project is a new Cauchy-Kubota type formula for convex functions which is a far-reaching generalization of a classical formula due to Cauchy that dates back to the 19th century. This formula not only allows for new representations of functional intrinsic volumes but also explains the possible singularities of the density functions that appear in their definitions. Next, a new functional version of the classical Steiner formula provides a new way to define functional intrinsic volumes as coefficients of a naturally arising polynomial and allows to describe these operators with the help of special mixed Monge-Ampère measures. Last but not least, the project's results can also explain why the initially conjectured inequalities do not hold true. Nonetheless, new Wulff-type inequalities for functional intrinsic volumes were found.