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.
