Convexity and Applications

Case Western Reserve University, Cleveland OH

Investigators

Abstract

The PI's research is in the two closely related areas of asymptotic geometric analysis and convex geometry, and their applications. Classical convexity studies the geometry of convex bodies in Euclidean space of a fixed dimension. Asymptotic geometric analysis deals with geometric properties of finite dimensional convex bodies as the dimension grows to infinity. In her research the PI uses methods from both areas, as well as probabilistic tools and concentration phenomena, to get a better understanding of the structure of convex sets. The PI was able to prove a variety of results where such structural aspects of convex sets play a role: in approximation of convex bodies by polytopes; to establish a link between the so called order statistics (which are fundamental objects in statistics) and Orlicz norms; to determine the ``sizes" of certain (convex) sets that appear naturally in quantum information theory. A further focus of the PI's research is the development of affine invariants. The PI and her collaborators started the systematic study of (affine invariant) functionals associated with convex bodies and their corresponding inequalities. Among the most important such functionals are affine surface area and p-affine surface area . The affine isoperimetric inequalities related to them are more powerful than their Euclidean relatives and related to other important inequalities, e.g. the Santalo- and Inverse Santalo- inequalities. The latter is related to Mahler's conjecture which is still open in dimension 3 and higher. A mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, if you want to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow us to predict its future motion from position and velocity we would need six separate numbers in all. If you want to track 100 distinct molecules of the air in the room then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases which are not visible in low dimensions. We can exploit those patters, thus taking advantage of the "curse of dimensionality" to make it the "blessing of dimensionality". It is one purpose of this grant to study such high dimensional phenomena.