Convexity and Applications

提案DMS-0305910 PI：Elisabeth Werner（凯斯西方）标题：凸性和应用摘要PI的研究是在经典凸性理论和凸几何分析。一个主要目标是得到一个更好的理解凸体的结构。为了做到这一点，她使用的技术从不同领域的数学：分析，微分几何，凸性理论，概率论。她研究等周不等式和仿射等周不等式。这些提供了强有力的工具，在表征和分类凸集。通过她的调查仿射表面积-最初是一个概念的仿射微分几何和发生在仿射等周不等式-她最近导致了广泛的研究问题的近似凸体的多面体。仿射表面积自然出现在这方面，因为它是relatedto边界结构的凸体。PI已经研究并仍在研究多面体逼近凸体的不同方面。例如，在一篇论文中，她和她的合作者一起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。除了凸性工具，概率工具，如浓度的措施，已被证明是非常有效的凸性。PI继续她的调查等概率结果推进她的研究在convexity结构的结果及其应用到本地Banach空间理论。过去的经验使PI相信纯理论的概念在应用中也是有价值的。她经历了这些领域的方法和结果在其他数学领域和应用领域中找到了应用：几何层析成像，一种起源于经典凸性理论的工具，给出了一种从其截面或投影中恢复凸形的方法。这被用于计算机视觉和图像分析，在生物学和医学中，凸形（器官）自然发生。几何算法在计算机科学中有着广泛的应用，经典凸性理论和几何分析的工具已经证明是量子信息论的有用工具。凸性理论与概率论、Banach空间理论、算子理论、新兴的随机矩阵理论、离散代数学的某些方向（包括复杂性理论问题、统计物理问题）、偏微分方程（包括由凸分析问题产生的非线性偏微分方程）等有着密切的联系。PI发现它非常刺激的互动与研究人员不仅从其他领域的数学，而且从应用领域。她已经与物理学家合作，并将继续这样做。特别是，她最近开始研究量子信息理论中的问题，其中凸性理论的方法非常有效。

ProposalDMS-0305910PI: Elisabeth Werner (Case Western)TITLE: CONVEXIT AND APPLICATIONSABSTRACT The PI's research is in classical convexity theory and convex geometric analysis.A primary goal is to get a better understanding of the structure of convex bodies.To do so she uses techniques from different areas of mathematics: analysis,differential geometry, convexity theory, probability theory.She investigates isoperimetric inequalities and affine isoperimetric inequalities.These provide powerful tools in characterizing and classifying convex sets.Through her investigation of the affine surface area -originally a concept of affinedifferential geometry and occurring in the affine isoperimetric inequality- she waslately led to an extensive study of questions of approximation of convex bodies bypolytopes.The affine surface area appears naturally in this context as it is relatedto the boundary structure of a convex body. The PI has investigated and still isinvestigating different aspects of approximation of convex bodies by polytopes. In one paper, for instance, she -together with her collaborator- proved the surprising result that random approximation by polytopes (choosing the vertices of theapproximating polytope randomly on the boundary of the body) is as good as bestapproximation. Besides convexity tools, probabilistic tools,like concentration ofmeasure, have proved to be very efficient in convexity. The PI continues her investigation of such probabilistic results for advancing her research in structuralresults in convexity and its applications to local Banach space theory. Past experience has led the PI to believe that purely theoretical concepts are alsouseful in applications. She has experienced that the methods and results from these areas find applications in other fields of mathematics and in applied areas:Geometric tomography, a tool having its origins in classical convexity theory,gives a method to recover convex shapes from its sections or projections. This isused in computer vision and image analysis, in biology and medicine where convexshapes(organs) occur naturally. Geometric algorithms find applications in computer science.Tools from classical convexity theory and geometric analysis have proved useful inquantum information theory. Convexity theory has mutually beneficialinteractions with probability theory, Banach space theory,operator theory, the newquickly developing theory of random matrices,some directions of discrete mathematicsincluding problems in complexity theory, problems of statistical physics,PDEs,including non-linear PDEs arising from problems in convex analysis. The PIfinds it very stimulating to interact with researchers not only from other areas of mathematics but also from applied areas. She has already worked with mathematicalphysicists and continues to do so. In particular, she has recently started to workon problems in quantum information theory where methods from convexity theory are very effective.