Convexity and Applications

This project is in asymptotic geometric analysis and affine convex geometry. One main emphasis of the research is on high dimensional objects and phenomena. This leads to applications in areas as physics, biology and medicine, computer science, optimization and economics and material science. Indeed, many objects appearing in these areas exhibit the property of being convex, including crystals, organs, high dimensional data clouds. How can one reconstruct or gain substantial knowledge of such convex objects when one only has partial information? This is one classical problem in convexity, called geometric tomography, and its research finds applications in medicine and biology where convex shapes occur naturally. In addition to the research activities, the PI will continue giving lectures to the scientific community, as well as to the general public and training graduate students and postdocs. Important features of the proposed 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 p-affine surface area. 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 PI 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, like e.g., dimension, the number of vertices of the approximating polytopes. These issues will be considered in Euclidean, spherical and hyperbolic space. The PI and her collaborators 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 spaces 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.