Mathematical Sciences: Isoperimetric Inequalities

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9507988 Lutwak The proposed research lies in the area of geometric convexity; more specifically, it focuses on the Brunn-Minkowski theory, otherwise known as the theory of mixed volumes. Various problems concerning convex bodies in space are proposed: the Minkowski problem, affine isoperimetric inequalities, and possible duals to the Brunn-Minkowski theory (replacing projections by intersections) are considered. The theory of mixed volumes has already found a wide variety of applications. In probability and statistics, it is used to supplement the Monte Carlo method. In geometric tomography, the theory is applied to various imaging (X-rays, and MRI scanners in particular) problems . One of the basic problems in the theory is the reconstruction of a convex body from a minimal set of its cross sections or projections.