Convexity and Applications

摘要奖：DMS-0072241首席研究员：Elizabeth Werner圆周社的研究涉及凸体的仿射几何和几何概率问题以及应用。仿射微分几何的经典定义可以追溯到Blaschke，它涉及光滑凸体的曲率函数，在等周不等式领域中有一个相当重要的工具。凸几何中的一个重要问题是将仿射表面积的运动推广到所有凸体。这个问题的解决是在最近十年里才完成的。目前，仿射曲面存在着许多延拓，其中有几个是PI发现的。在这些推广过程中发展的新技术和新思想对其他问题应该是有益的，PI建议将这些技术应用于用多面体逼近凸体的问题。这些逼近问题得到了深入的研究，并在数学和计算机科学的许多领域得到了应用。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，即多面体的随机逼近(在物体边界上随机选择逼近多面体的顶点)与最佳逼近一样好。概率统计中的高斯相关猜想认为，在标准的高斯度量下，原点对称凸集是正相关的。尽管许多研究人员在过去几年里得出了几个部分结果，包括PI，但这个猜想仍然没有定论。PI和她的合作者在此背景下获得的对高斯分布尾部的最佳估计对于数学物理中的问题也是相关的(例如，参见《微分方程式和数学物理》，国际出版社，2000年，第43-51页)。PI想要了解凸集的结构。为了做到这一点，她使用了不同数学领域的技术：分析、微分几何、凸性理论。人们想要了解这种集合的结构，因为它们不仅自然地出现在数学和数学物理的其他分支中，而且还出现在应用领域、象形学和图像分析以及计算机科学中。PI和她的合作者计划继续研究凸性和其他更适用于数学的领域的问题。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中的工具。

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.