Mathematical Sciences: Banach Space Theory and Convexity Theory

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9401784 Werner Professor Werner proposes to continue her research in Banach space theory and convexity theory. In convexity theory she wants to investigate further properties of floating and illumination bodies of convex bodies as well as their connection with affine surface area and approximation of convex bodies by polytopes. In Banach space theory she plans to continue her work on problems related to the Radon Nikodym property. A solid body is called convex in case a line segment joining two of its points stays within the body. Convex bodies can frequently be described as intersections of nice convex sets such as half-planes. One of the questions considered here is whether a convex body can be nicely approximated by convex bodies having only a finite number of vertices. The approximation is to be good in the sense that volumes of the approximating bodies are close to the volume of the original body or in the sense that their surface areas are close. The second project involves abstract mathematics that has applications to probability theory. ***