Fixed point problems and isoperimetric inequalities

The Brunn-Minkowski theory of convex bodies has emerged from the study of the interaction of two basic notions in geometry: vector addition and volume. One of the primary goals of the theory is the study of isoperimetric type problems for convex bodies. The simplest and most fundamental examples of such problems is the classic isoperimetric problem that relates the volume of a body with its surface area: among bodies of a given volume, the Euclidean ball is the one with minimal surface area. This classic example can be generalized in many ways if one replaces volume and surface area by other global geometric invariants. In this project, we will consider geometric invariants that are defined in terms of the volume of the images of convex bodies under certain geometric operators: Minkowski valuations. Using varational methods, we will reduce the search of possible extremizers of these isoperimetric type problems to solutions of fixed-point equations of the underlining geometric operators. We will also consider extensions of our problems to the Lp Brunn-Minkowski theory which fundamentally extends the classical Brunn-Minkowski theory and which, due to its rapid development, is now established as one of the centers of research topics in Convex Geometric Analysis. We will treat some other emerging research trends in the complex regime for which we will also study some versions of these isoperimetric problems for complex convex bodies.

The Brunn-Minkowski theory of convex bodies has emerged from the study of the interaction of two basic notions in geometry: vector addition and volume. One of the primary goals of the theory is the study of isoperimetric-type problems for convex bodies. The simplest and most fundamental example of such problems is the classic isoperimetric problem that relates the volume of a body with its surface area: among bodies of a given volume, the Euclidean ball is the one with minimal surface area. This classic example can be generalized in many ways if one replaces volume and surface area with other global geometric invariants. In this project, we will consider geometric invariants that are defined in terms of the volume of the images of convex bodies under certain geometric operators: Minkowski valuations. Using variational methods, we will reduce the search for possible extremizers of these isoperimetric-type problems to solutions of fixed-point equations of the underlying geometric operators. We will also consider extensions of our problems to the L_p Brunn-Minkowski theory, which fundamentally extends the classical Brunn-Minkowski theory and which, due to its rapid development, is now established as one of the central research topics in convex geometric analysis. We will treat some other emerging research trends in the complex regime, for which we will also study some versions of these isoperimetric problems for complex convex bodies.