Mathematical Sciences: Mixed Volumes

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9626688 Klain This proposal lies in the area of convexity and integral geometry. The investigator plans to work on the theory of mixed volumes with a view towards characterizing valuations (valuations include all elementary mixed volumes) and set functions invariant under various group actions. This is an extension of Hadwiger's theorem for the elementary mixed volumes. Hadwiger's theorem classifies all continuous valuations on convex bodies that are invariant under rotations and translations. A central question in integral geometry is to find a formula for the volumes of various combinations of a given set of convex bodies whose volumes are already known. Coefficients arising in such a formula are called mixed volumes. The proposed research attempts to compute mixed volumes in various special cases; consider applications to probability theory and integral geometry.