Mathematical Sciences: Computational Aspects of Some Problems in Convex Geometry

研究人员Lopez 9626749和他的同事从两个方面研究了凸几何的计算方面：(1)设计和实现了用具有先验指定顶点数的凸多面体逼近多维空间中凸体的高效算法。在不同的应用领域中出现的具体实例，例如较小的多面体对多面体的逼近，给予了特别的关注。扩展，如在各种度量下的逼近，也被考虑。(2)应用高性能计算来解决或向解决凸几何中的一些长期存在的问题迈进。一个重要的例子是马勒关于凸体及其极体的体积积的猜想。通过将搜索空间缩减为有限但大量的可通过计算验证的情况集来解决该问题的一个实例，并将其应用于小波理论。这位研究人员和他的同事研究了用指定大小的多面体(由平面形成的实体)逼近多维凸体问题的有效计算解决方案。这种近似在许多学科中都是一个重要的工具，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，多面体是迄今为止使用最广泛的模型表示形式。因此，这项工作也很重要，因为它促进了对多面体已经可用的大量方法的使用，前提是所得到的近似是好的，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。一种基于对大量但有限数量的可能情况进行计算验证的方法有望在该领域产生重要的进步。

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.