Mathematical Sciences: Mixed Volumes

数学科学：混合卷

• 批准号: 9626688
• 负责人: Daniel Klain
• 金额: $ 4万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626688&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Mixed; Volumes

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.

小行星9626688 这个建议是在凸性和积分几何领域。研究人员计划研究混合体积理论，以期表征估值（估值包括所有基本混合体积）和在各种群作用下不变的集合函数。这是Hadwiger定理在初等混合体积中的推广。Hadwiger定理分类了凸体上所有在旋转和平移下不变的连续赋值。 积分几何中的一个中心问题是找到一个公式的各种组合的凸体的体积已经知道。在这样的公式中出现的系数称为混合体积。拟议的研究试图计算混合体积在各种特殊情况下，考虑应用概率论和积分几何。