Convexity and Applications

凸性及其应用

• 批准号: 0072241
• 负责人: Elisabeth Werner
• 金额: $ 9.29万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072241&HistoricalAwards=false
• 关键词: Convexity; Applications

AbstractAward: DMS-0072241Principal Investigator: Elizabeth WernerThe PI's research deals with questions in affine geometry andgeometric probability of convex bodies as well as withapplications. A tool of considerable importance in the area ofisoperimetric inequalities is the affine surface area from affinedifferential geometry whose classical definition goes back toBlaschke and involves the curvature function of a smooth convexbody. An important problem in convex geometry was to extend thenotion of affine surface area to all convex bodies. The solutionof this problem has only been completed within the lastdecade. At present, many extensions of the affine surface areaexist, several of them discovered by the PI. The new techniquesand ideas developed in the process of these extensions should bebeneficial for other problems, and the PI proposes to apply thesetechniques to problems of approximation of convex bodies bypolytopes. These approximation problems have been studiedextensively and find application in many areas of mathematics andcomputer science. In one paper, for instance, the PI and hercollaborator proved the surprising result that randomapproximation by polytopes (choosing the vertices of theapproximating polytope randomly on the boundary of the body) isas good as best approximation. The Gaussian correlationconjecture in probability and statistics asserts thatorigin-symmetric convex sets are positively correlated under thestandard Gaussian measure. In spite of several partial resultsobtained by many researchers within the last years, including thePI, the conjecture remains undecided. Optimal estimates for thetail of the Gaussian distribution obtained in this context by thePI and her collaborators are also relevant for problems inmathematical physics (see e.g.Differential Equations andMathematical Physics, International Press 2000, p.43-51).The PI wants to get an understanding of the structure of convexsets. To do so she uses techniques from different areas ofmathematics: analysis, differential geometry, convexitytheory. One wants to understand the structure of such sets asthey appear naturally not only in other branches of mathematicsand mathematical physics, but also in applied areas, liketomography and image analysis, and computer sciences. The PI andher collaborators plan to continue working on problems whereconvexity and other areas of more applied mathematicsinteract. Currently she is involved in a project related toquantum computing which uses -among other things- tools fromconvexity theory.

AbstractAward：DMS-0072241首席研究员：Elizabeth Werner PI的研究涉及仿射几何和凸体的几何概率以及应用。 一个工具相当重要的领域ofisoperimetric不等式是仿射表面积从仿射微分几何的经典定义可以追溯到Blaschke和涉及曲率函数的光滑凸体。凸几何中的一个重要问题是将仿射表面积的概念推广到所有凸体。这一问题的解决是近十年来才完成的。目前，仿射曲面的许多扩展区域存在，其中一些是由PI发现的。在这些扩展过程中开发的新技术和思想应该对其他问题有益，PI建议将这些技术应用于凸体的多面体逼近问题。这些近似问题在数学和计算机科学的许多领域都得到了广泛的研究和应用。例如，在一篇论文中，PI和她的合作者证明了一个令人惊讶的结果，即多面体的随机近似（在物体的边界上随机选择近似多面体的顶点）与最佳近似一样好。 概率统计中的高斯相关性猜想认为，在标准高斯测度下，源对称凸集是正相关的。 尽管在过去的几年里，包括PI在内的许多研究人员都得到了一些部分结果，但该猜想仍然悬而未决。 PI和她的合作者在此背景下获得的高斯分布尾部的最佳估计也与数学物理问题有关（参见例如微分方程和数学物理，国际出版社2000年，第43 -51页）。PI希望了解凸集的结构。 为了做到这一点，她使用了来自不同数学领域的技术：分析，微分几何，凸性理论。人们希望了解这些集合的结构，因为它们不仅在数学和数学物理的其他分支中自然出现，而且在应用领域，如断层扫描和图像分析以及计算机科学中也是如此。 PI和她的合作者计划继续研究凸性和其他更多应用几何学领域相互作用的问题。目前，她参与了一个与量子计算有关的项目，该项目使用了凸性理论的工具。