Isoperimetric Inequalities

本文研究的是几何凸性的解析部分。更具体地说，它侧重于布伦-闵可夫斯基理论，也被称为混合体积理论。主要研究人员计划将理论推广到比凸体更一般的物体上，并以Lp空间理论扩展L1和L2空间理论的相同方式扩展理论。他们还将继续致力于创建布伦-闵可夫斯基理论的对偶，其中交集取代了投影。在此基础上，他们打算研究在这个扩展的布伦-闵可夫斯基理论中产生的偏微分方程，特别是对经典闵可夫斯基问题的推广。这项工作的核心部分是在扩展的布伦-闵可夫斯基理论及其对偶中建立新的等周不等式。混合体积理论已经有了广泛的应用，从统计学到几何层析成像。（几何层析成像是一门试图从低维信息（如x射线、投影或所讨论物体的部分）描述三维物体的学科。）主要研究人员打算扩展和建立这一重要数学理论的新的类似物。这项工作的核心部分将涉及新的等周不等式的发明（或发现）。这些不等式，特别是仿射等周不等式，已被证明在从偏微分方程到几何断层扫描到图像分析等学科中都很重要。它们非常有用，因为它们提供了一个量的估计，比如一个物体的体积，而已知的只有另一个量，比如物体的表面积。

The proposed research lies in the analytic part of geometric convexity. More specifically, it focuses on the Brunn-Minkowski theory, also known as the theory of mixed volumes. The principal investigators plan to generalize the theory to objects more general than convex bodies and extend the theory in the same way that the theory of Lp spaces extends the theory of L1 and L2 spaces. They will also continue work on the creation of a dual of the Brunn-Minkowski theory, where intersections replace projections. Following on this, they intend to study the partial differential equations which arise in this extended Brunn-Minkowski theory, specifically generalizations of the classical Minkowski problem. A central part of this work is establishing new isoperimetric inequalities within the extended Brunn-Minkowski theory and its dual. The theory of mixed volumes has already found a wide variety of applications ranging from statistics to geometric tomography. (Geometric tomography is the subject which attempts to describe three-dimensional objects from lower dimensional information such as X-rays, projections, or sections of the objects in question.) The principal investigators intend to extend and build new analogs of this important mathematical theory. A central part of this work will involve the invention (or discovery) of new isoperimetric inequalities. These inequalities, in particular affine isoperimetric inequalities, have proven to be of importance in subjects ranging from partial differential equations to geometric tomography to image analysis. They are very useful because they provide estimates about how large one quantity can be, say the volume of a body, when all that is known is some other quantity, say the surface area of the body.