Convex valuation theory and integral geometry

凸估价理论和积分几何

• 批准号: RGPIN-2016-06764
• 负责人: Faifman, Dmitry
• 金额: $ 1.15万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=616502
• 关键词: Convex; valuation; theory; integral; geometry

Convex geometry is a classical, yet rapidly developing branch of mathematics, appearing in various guises in a great variety of mathematical disciplines, ranging from algebraic and symplectic geometry, through probability theory and combinatorics, and on to algorithmics. Integral geometry appeared in the nineteenth century in several flavors. It distinguishes itself from differential geometry by studying global invariants (such as total length), rather than local invariants (such as curvature). My research concerns valuation theory. Valuation theory started with Dehn's solution of Hilbert's third problem on the definition of volume of polytopes in 3-dimensional space. In the simplest setting, valuations are continuously varying, finitely-additive measures on convex bodies. Many of the integral-geometric invariants of convex bodies, such as volume and surface area, are in fact valuations. Today, valuation theory is central in convex geometry, integral geometry, stochastic geometry and geometric probability. A natural problem in valuation theory is to find all valuations that are invariant under a group of symmetries. For example, volume and surface area remain unchanged during translations and rotations of a body. When the isotropy group of symmetries is compact, such as the group of rotations, a large toolbox, developed over the past 15 years, is readily available for the classification problem. I am interested primarily in non-compact groups of symmetries, where new approaches must be developed. The famous isoperimetric inequality has a far reaching generalization, the Alexandrov-Fenchel inequality, which concerns valuations of a particular kind known as mixed volumes. It is therefore natural to look for other geometric inequalities involving valuations. Another central player in integral geometry is the Radon transform, which nominally replaces a function by its integrals over lines, and forms the basis of the field of tomography, used in medical imaging, seismography etc. Valuation theory offers a vast generalization of the Radon transform, which I plan to explore.

凸几何是一个经典的，但迅速发展的数学分支，出现在各种各样的数学学科，从代数和辛几何，通过概率论和组合，并在算法的各种伪装。积分几何出现在十九世纪的几个风味。它与微分几何的区别在于研究全局不变量（如总长度），而不是局部不变量（如曲率）。 我的研究涉及估值理论。赋值理论开始于德恩对希尔伯特第三个问题的解决，这个问题是关于三维空间中多面体体积的定义。在最简单的设置中，赋值是凸体上连续变化的有限可加测度。凸体的许多积分几何不变量，如体积和表面积，实际上是赋值。今天，赋值理论是凸几何、积分几何、随机几何和几何概率的核心。 赋值理论中的一个自然问题是找到在一组对称下不变的所有赋值。例如，体积和表面积在物体的平移和旋转期间保持不变。当对称性的各向同性组是紧凑的，如旋转组，一个大的工具箱，在过去的15年中开发的，是现成的分类问题。我感兴趣的主要是非紧凑组的对称性，新的方法必须制定。 著名的等周不等式有一个意义深远的推广，即亚历山德罗夫-芬切尔不等式，它涉及一种特殊类型的估值，称为混合体积。因此，寻找其他涉及赋值的几何不等式是很自然的。 积分几何中的另一个核心角色是Radon变换，它名义上用它在线上的积分来代替函数，并形成了断层摄影领域的基础，用于医学成像，地震学等。