Principal Investigator: Elizabeth Werner
The PI's research deals with questions in affine geometry and geometric probability of convex bodies as well as with applications. A tool of considerable importance in the area of isoperimetric inequalities is the affine surface area from affine differential geometry whose classical definition goes back to Blaschke and involves the curvature function of a smooth convex body. An important problem in convex geometry was to extend the notion of affine surface area to all convex bodies. The solution of this problem has only been completed within the last decade. At present, many extensions of the affine surface area exist, several of them discovered by the PI. The new techniques and ideas developed in the process of these extensions should be beneficial for other problems, and the PI proposes to apply these techniques to problems of approximation of convex bodies by polytopes. These approximation problems have been studied extensively and find application in many areas of mathematics and computer science. In one paper, for instance, the PI and her collaborator proved the surprising result that random approximation by polytopes (choosing the vertices of the approximating polytope randomly on the boundary of the body) is as good as best approximation. The Gaussian correlation conjecture in probability and statistics asserts that origin-symmetric convex sets are positively correlated under the standard Gaussian measure. In spite of several partial results obtained by many researchers within the last years, including the PI, the conjecture remains undecided. Optimal estimates for the tail of the Gaussian distribution obtained in this context by the PI and her collaborators are also relevant for problems in mathematical physics (see e.g.Differential Equations and Mathematical Physics, International Press 2000, p.43-51).
The PI wants to get an understanding of the structure of convex sets. To do so she uses techniques from different areas of mathematics: analysis, differential geometry, convexity theory. One wants to understand the structure of such sets as they appear naturally not only in other branches of mathematics and mathematical physics, but also in applied areas, like tomography and image analysis, and computer sciences. The PI and her collaborators plan to continue working on problems where convexity and other areas of more applied mathematics interact. Currently she is involved in a project related to quantum computing which uses -among other things- tools from convexity theory.
