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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=650397
• 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.**

凸几何是一个经典但发展迅速的数学分支，以各种形式出现在各种数学学科中，从代数几何和辛几何，到概率论和组合学，再到算法学。十九世纪的积分几何以多种形式出现。它通过研究全局不变量（例如总长度）而不是局部不变量（例如曲率）来区别于微分几何。*** ***我的研究涉及估值理论。估价理论始于德恩对希尔伯特第三个问题的解决，即三维空间中多胞体体积的定义。在最简单的环境中，估值是凸体上不断变化的、有限可加的度量。凸体的许多积分几何不变量，例如体积和表面积，实际上是估值。如今，估值理论是凸几何、积分几何、随机几何和几何概率的核心。************估值理论中的一个自然问题是找到一组对称性下所有不变的估值。例如，在物体平移和旋转期间，体积和表面积保持不变。当各向同性对称群紧致时，例如旋转群，过去 15 年开发的大型工具箱可随时用于分类问题。我主要对非紧对称群感兴趣，必须开发新的方法。 *********著名的等周不等式具有深远的概括性，即亚历山德罗夫-芬切尔不等式，它涉及一种称为混合体的特殊类型的估值。因此，寻找涉及估值的其他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，它名义上用函数在直线上的积分来代替函数，并构成断层扫描领域的基础，用于医学成像、地震学等。估值理论提供了 Radon 变换的广泛概括，我计划对此进行探索。 **