Convexity and Applications

本文主要研究经典凸性理论和凸几何分析。主要目的是更好地理解凸体的结构。为此，她运用了不同数学领域的方法：分析、微分几何、凸性理论和凸性理论。概率论。她研究等周不等式和仿射等周不等式，这为刻画凸集和对凸集分类提供了强有力的工具。通过她对仿射表面积的研究--最初是仿射微分几何的概念，出现在仿射等周不等式中--她最近引起了对凸体用多边形逼近问题的广泛研究。仿射表面积在这种情况下自然出现，因为它与凸体的边界结构有关。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，她和她的合作者证明了一个令人惊讶的结果，即多面体的随机逼近(在身体边界上随机选择逼近多面体的顶点)与最佳逼近一样好。除了凸性工具外，概率工具，如度量的集中度，已被证明在凸性方面非常有效。PI继续研究这些概率结果，以推进她在凸性的结构结果及其在局部Banach空间理论中的应用的研究。过去的经验使PI相信纯粹的理论概念在应用中也是有用的。她经历过，这些领域的方法和结果在其他数学领域和应用领域得到了应用：几何层析成像是一种起源于经典凸性理论的工具，它提供了一种从截面或投影中恢复凸面形状的方法。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)是自然产生的。几何算法在计算机科学中得到了应用，经典凸性理论和几何分析的工具已被证明是有用的量子信息论。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、离散数学的一些方向，包括复杂性理论中的问题、统计物理问题、偏微分方程组，包括由凸分析中的问题产生的非线性偏微分方程组，都有着互惠互利的相互作用。PI发现，不仅与其他数学领域的研究人员互动，而且与应用领域的研究人员互动，也非常令人兴奋。她已经与数学物理学家合作过，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，在这些问题中，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.