Convexity and Applications

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ProposalDMS-0305910PI: Elisabeth Werner (Case Western)TITLE: CONVEXIT AND APPLICATIONSABSTRACT The PI's research is in classical convexity theory and convex geometric analysis.A primary goal is to get a better understanding of the structure of convex bodies.To do so she uses techniques from different areas of mathematics: analysis,differential geometry, convexity theory, probability theory.She investigates isoperimetric inequalities and affine isoperimetric inequalities.These provide powerful tools in characterizing and classifying convex sets.Through her investigation of the affine surface area -originally a concept of affinedifferential geometry and occurring in the affine isoperimetric inequality- she waslately led to an extensive study of questions of approximation of convex bodies bypolytopes.The affine surface area appears naturally in this context as it is relatedto the boundary structure of a convex body. The PI has investigated and still isinvestigating different aspects of approximation of convex bodies by polytopes. In one paper, for instance, she -together with her collaborator- proved the surprising result that random approximation by polytopes (choosing the vertices of theapproximating polytope randomly on the boundary of the body) is as good as bestapproximation. Besides convexity tools, probabilistic tools,like concentration ofmeasure, have proved to be very efficient in convexity. The PI continues her investigation of such probabilistic results for advancing her research in structuralresults in convexity and its applications to local Banach space theory. Past experience has led the PI to believe that purely theoretical concepts are alsouseful in applications. She has experienced that the methods and results from these areas find applications in other fields of mathematics and in applied areas:Geometric tomography, a tool having its origins in classical convexity theory,gives a method to recover convex shapes from its sections or projections. This isused in computer vision and image analysis, in biology and medicine where convexshapes(organs) occur naturally. Geometric algorithms find applications in computer science.Tools from classical convexity theory and geometric analysis have proved useful inquantum information theory. Convexity theory has mutually beneficialinteractions with probability theory, Banach space theory,operator theory, the newquickly developing theory of random matrices,some directions of discrete mathematicsincluding problems in complexity theory, problems of statistical physics,PDEs,including non-linear PDEs arising from problems in convex analysis. The PIfinds it very stimulating to interact with researchers not only from other areas of mathematics but also from applied areas. She has already worked with mathematicalphysicists and continues to do so. In particular, she has recently started to workon problems in quantum information theory where methods from convexity theory are very effective.