Mathematical Sciences: Mixed Volumes

9626688克莱恩这个建议涉及到凸性和积分几何领域。研究人员计划研究混合体理论，以刻画赋值(赋值包括所有基本的混合体)和集函数在各种群作用下不变的特征。这是Hadwiger关于初等混合体的定理的推广。Hadwiger定理对凸体上所有在旋转和平移下不变的连续赋值进行了分类。积分几何中的一个中心问题是求出一组已知体积的凸体的各种组合的体积的公式。在这样的公式中出现的系数称为混合体积。建议的研究试图在各种特殊情况下计算混合体积；考虑在概率论和积分几何中的应用。

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.