This research is concerned with developing efficient algorithms and paradigms to solve geometric problems for digitized pictures on hypercube multiprocessors. At present, it appears that commercially available medium-grained hypercube multiprocessors are not well suited to low level vision tasks, such as convolution and Hough transform. Therefore, our research has focused on medium level vision problems involving connectivity, proximity, and convexity. In this paper, data reduction techniques are developed for medium level vision tasks. These techniques are used to present efficient hypercube algorithms for solving the convex hull problem. Results are given for implementing a variety of convex hull algorithms on an Intel iPSC1 hypercube. Implementation issues and algorithm paradigms are discussed in their relationship to the running times of the algorithms on this machine.