Mathematical Sciences: Computational Aspects of Some Problems in Convex Geometry

研究者和他的同事从两个方向研究凸几何的计算方面：(1)设计和实现通过具有先验指定顶点数量的凸多边形在多维空间中逼近凸体的有效算法。在各种应用领域中出现的具体实例，例如用较小的多面体逼近多面体，都得到了特别的注意。扩展，如在各种度量下的近似，也被考虑。(2)应用高性能计算解决或朝着解决凸几何中许多长期存在的问题的方向发展。一个重要的例子是马勒关于凸体与极体体积积的猜想。这个问题的一个应用于小波理论的实例是通过将搜索空间减少到一个有限但可以计算验证的大的情况集来解决的。研究者及其同事研究了用规定尺寸的多面体（由平面形成的固体）近似多维凸体问题的有效计算解决方案。这种近似是许多学科的重要工具，包括分子建模、最优控制、计算机辅助设计、模式识别和计算机可视化。此外，由于其简单性，多面体是迄今为止最广泛使用的模型表示形式。因此，这项工作也很重要，因为它促进了对多面体已有的大量方法的使用，只要得到的近似是好的，并且可以有效地执行。具有象征意义的是，使用高性能计算工具来解决长期存在的几何问题。一种基于计算验证大量但有限数量的可能情况的方法有望在该领域取得重要进展。

Lopez 9626749 The investigator and his colleague study computational aspects of convex geometry in two directions: (1) Design and implementation of efficient algorithms for the approximation of convex bodies in multidimensional space by convex polytopes with a priori specified number of vertices. Specific instances that arise in various application areas, such as the approximation of polytopes by smaller polytopes, are given special attention. Extensions, such as approximation under various metrics, also are considered. (2) Application of high performance computing to solve or to advance toward the solution of a number of long-standing problems in convex geometry. An important example is Mahler's conjecture on the volume-product of a convex body and its polar body. An instance of this problem with applications to the theory of wavelets is tackled by reducing the search space to a finite but large set of cases that can be verified computationally. The investigator and colleague study efficient computational solutions for the problem of approximating multidimensional convex bodies by polytopes (solids formed by plane faces) of a prescribed size. Such approximation is an important tool in many disciplines, including molecular modeling, optimal control, computer-aided design, pattern recognition, and computer visualization. Furthermore, due to their simplicity, polytopes are by far the most widely used form of model representation. Thus, the work is also important because it facilitates the use of the large body of methods already available for polytopes, provided that the resulting approximation is good and can be performed efficiently. Symbolically, long-standing geometric problems are attacked using the tools of high performance computing. An approach based on verifying computationally a large but finite number of possible cases promises to yield an important advance in the field.