On the Partition of Space by Hyperplanes

Abstract

We consider the problem of partitioning of space by hyperplanes that arises in many application areas, where the number of regions the space is divided into is required to be determined, such as speech/pattern recognition, various classification problems, data analysis. We obtain some relations for the number of divisions and establish a recurrence relation for the maximum number of regions in d-dimensional Euclidean space cut by n hyperplanes. We also re-derive an explicit formula for the number of regions into which the space can be partitioned by n hyperplanes.