Convexity and Applications

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A major emphasis of this project is on high dimensional objects and phenomena. These appear in areas as diverse as physics, biology and medicine, computer science, optimization, economics, and material sciences. A mathematical description of a scientific or engineering question often requires many 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 the difficulty of sampling, and computation goes up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases that are not visible in low dimensions. We can exploit those patterns thus taking advantage of the "curse of dimensionality" to make it the "blessing of dimensionality". It is one purpose of these projects to study such high dimensional phenomena.These projects are in asymptotic geometric analysis and affine convex geometry. 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 principal investigator 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. 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. This award includes support for a graduate student.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.