Mathematical Sciences: Isoperimetric Inequalities

首席研究员将解决等距不平等领域的一些问题。这些问题包括广义凸超曲面，仿射面积的不等式，交集和投影问题，以及对偶Brunn-Minkowski理论。广义凸体与凸体的关系，就像广义函数(分布)与普通函数的关系一样。它们的应用将使首席研究者能够建立一些猜想的仿射等距不等式。经典的Brunn-Minkowski理论非常适合处理涉及凸形投影的问题。在一些重要问题上进展缓慢是因为这个理论不适合处理相交问题。首席研究员将使用对偶Brunn-Minkowski理论来克服这些困难。这项工作在数学以外的领域也有应用，比如体视学。

The principal investigator will solve a number of problems in the area of isometric inequalities. These involve generalized convex hypersurfaces, inequalities for affine surface area, intersection and projection problems, and the dual Brunn- Minkowski theory. Generalized convex bodies are related to convex bodies in the way generalized functions (distributions) are related to ordinary functions. Their utilization will enable the principal investigator to establish a number of conjectured affine isometric inequalities. The classical Brunn-Minkowski theory is ideal for handling questions involving projections of convex shapes. The slow progress on a number of important questions is due to the fact that this theory is unsuited for dealing with intersection problems. The principal investigator will use a dual Brunn- Minkowski theory to overcome these difficulties. This work has applications outside mathematics to such fields as stereology.