Mathematical Sciences: Computational Aspects of Some Problems in Convex Geometry

洛佩斯9626749 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有效算法。 在各种应用领域中出现的具体实例，如由较小的多面体近似多面体，给予特别关注。 扩展，如近似下的各种指标，也被认为是。(2)应用高性能计算来解决或推进凸几何中一些长期存在的问题。 一个重要的例子是马勒关于凸体及其极体的积的猜想。 一个例子，这个问题的应用小波理论的解决，减少搜索空间，以有限的，但大的一组情况下，可以通过计算验证。 调查员和同事研究有效的计算解决方案的问题，近似多维凸体的多面体（固体形成的平面）的规定大小。 这种近似是许多学科中的重要工具，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，多面体是迄今为止使用最广泛的模型表示形式。 因此，这项工作也是重要的，因为它有利于使用大量的方法已经可用于多面体，提供了所得到的近似是好的，可以有效地执行。 象征性地，长期存在的几何问题使用高性能计算的工具进行攻击。 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.