Convexity and Applications

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AbstractAward: DMS-0072241Principal Investigator: Elizabeth WernerThe PI's research deals with questions in affine geometry andgeometric probability of convex bodies as well as withapplications. A tool of considerable importance in the area ofisoperimetric inequalities is the affine surface area from affinedifferential geometry whose classical definition goes back toBlaschke and involves the curvature function of a smooth convexbody. An important problem in convex geometry was to extend thenotion of affine surface area to all convex bodies. The solutionof this problem has only been completed within the lastdecade. At present, many extensions of the affine surface areaexist, several of them discovered by the PI. The new techniquesand ideas developed in the process of these extensions should bebeneficial for other problems, and the PI proposes to apply thesetechniques to problems of approximation of convex bodies bypolytopes. These approximation problems have been studiedextensively and find application in many areas of mathematics andcomputer science. In one paper, for instance, the PI and hercollaborator proved the surprising result that randomapproximation by polytopes (choosing the vertices of theapproximating polytope randomly on the boundary of the body) isas good as best approximation. The Gaussian correlationconjecture in probability and statistics asserts thatorigin-symmetric convex sets are positively correlated under thestandard Gaussian measure. In spite of several partial resultsobtained by many researchers within the last years, including thePI, the conjecture remains undecided. Optimal estimates for thetail of the Gaussian distribution obtained in this context by thePI and her collaborators are also relevant for problems inmathematical physics (see e.g.Differential Equations andMathematical Physics, International Press 2000, p.43-51).The PI wants to get an understanding of the structure of convexsets. To do so she uses techniques from different areas ofmathematics: analysis, differential geometry, convexitytheory. One wants to understand the structure of such sets asthey appear naturally not only in other branches of mathematicsand mathematical physics, but also in applied areas, liketomography and image analysis, and computer sciences. The PI andher collaborators plan to continue working on problems whereconvexity and other areas of more applied mathematicsinteract. Currently she is involved in a project related toquantum computing which uses -among other things- tools fromconvexity theory.