Mathematical Sciences: Isoperimetric Inequalities

数学科学：等周不等式

• 批准号: 9507988
• 负责人: Erwin Lutwak
• 金额: $ 9.08万
• 依托单位: Polytechnic University of New York
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1998-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9507988&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Isoperimetric; Inequalities

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.

9507988卢特瓦克（Lutwak）提出的研究在于几何凸面。更具体地说，它的重点是Brunn-Minkowski理论，也称为混合体积理论。提出了有关空间中有关凸体的各种问题：Minkowski问题，仿射等级不平等以及可能对Brunn-Minkowski理论（用交叉路口替换投影）可能的双重问题。 混合量的理论已经发现了各种各样的应用。在概率和统计中，它用于补充蒙特卡洛法。在几何断层扫描中，该理论应用于各种成像（尤其是X射线和MRI扫描仪）问题。该理论中的基本问题之一是从最小的横截面或投影组中重建了凸体。