Isoperimetric Inequalities

所提出的研究在于几何凸性的解析部分。 更具体地说，它侧重于Brunn-Minkowski理论，也被称为混合体积理论。主要研究人员计划将该理论推广到比凸体更一般的对象，并以Lp空间理论扩展L1和L2空间理论的相同方式扩展该理论。 他们还将继续致力于创建Brunn-Minkowski理论的对偶，其中交叉取代投影。 在此之后，他们打算研究偏微分方程中出现的扩展Brunn-Minkowski理论，特别是推广的经典Minkowski问题。这项工作的一个核心部分是建立新的等周不等式内的扩展Brunn-Minkowski理论及其对偶。 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。（几何层析成像是试图从低维信息（如X射线、投影或物体的截面）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，已被证明是重要的主题，从偏微分方程几何层析成像图像分析。它们非常有用，因为它们提供了一个量（比如说物体的体积）有多大的估计，而我们所知道的只是另一个量（比如说物体的表面积）。

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.