Convexity and Applications

Case Western Reserve University, Cleveland OH

Investigators

Abstract

The principal investigator's research is in asymptotic geometric analysis and affine convex geometry. One main emphasis of her research is on high-dimensional objects and phenomena. This leads to applications of her research in areas as diverse as physics, biology and medicine, computer science, optimization and economics and material science: Indeed, 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, specifying the location, direction and speed of one gas molecule in a room 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 data scientists 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 patterns, thus converting the "curse of dimensionality" into "blessing of dimensionality". It is one purpose of this award to study such high-dimensional phenomena. Important features of this project are the study of high-dimensional objects and phenomena and their links with other areas of mathematics and mathematical sciences, such as probability, statistics and information theory. Of particular interest are the affine invariant functionals on convex bodies in high dimensions. Among the most important such functionals are affine surface area and the p-affine surface area (a family of functionals parametrized by a real number p). Their corresponding affine isoperimetric inequalities, established by the PI and collaborators for all p, are stronger than their Euclidean counterparts and related to the famous Mahler conjecture which is still open in dimensions four and higher. It was shown by the principal investigator that p-affine surface areas are directly related to entropies of cone measures of convex bodies which establishes a link between convex geometry and information theory. This link will be further explored, also in the context of log concave functions which are a natural extension of convex bodies in the realm of functions. Moreover, affine surface area appears naturally in questions on approximation of convex bodies by polytopes, a further main topic of study. The goal is to establish optimal dependence on all the relevant parameters involved in the approximation, for example the dimension and the number of vertices of the approximating polytopes. The principal investigator and her collaborators also extended the notions of affine surface area recently to a functional setting and to spherical and hyperbolic space. To establish the corresponding inequalities in those settings is a further topic of study. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.