Convexity and Applications

主要研究者：Elizabeth wernerpi的研究涉及仿射几何和凸体的几何概率问题及其应用。在等周不等式领域中，一个相当重要的工具是仿射微分几何中的仿射表面积，其经典定义可以追溯到blaschke，并涉及光滑凸体的曲率函数。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。这个问题的解决直到最近十年才完成。目前，存在许多仿射表面积的扩展，其中一些是由PI发现的。在这些扩展过程中产生的新技术和新思想对其他问题也有一定的借鉴意义，并建议将这些技术应用于凸体多面体的逼近问题。这些近似问题已经得到了广泛的研究，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（在物体的边界上随机选择逼近多面体的顶点）与最佳逼近一样好。概率论和统计学中的高斯相关猜想断言原点对称凸集在标准高斯测度下是正相关的。尽管包括pi在内的许多研究人员在过去几年中获得了一些部分结果，但这个猜想仍然悬而未决。在这种情况下，由pi和她的合作者获得的高斯分布尾部的最佳估计也与数学物理问题相关（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、凸理论。人们想要理解这些集合的结构，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。PI和她的合作者计划继续研究凹性和其他应用数学领域相互作用的问题。目前，她正在参与一个与量子计算相关的项目，该项目使用了来自凸性理论的工具。

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.