subarrangements

A subarrangement is a concept primarily found in computational geometry and combinatorics, referring to a specific type of subdivision of geometric space. Given a set of hyperplanes in a real vector space, the arrangement of these hyperplanes partitions the space into convex regions. A subarrangement is formed by considering only a subset of these original hyperplanes. The regions in a subarrangement are therefore larger than or equal to the regions in the full arrangement, as fewer constraints are imposed on the space.

The structure of subarrangements is intimately related to the structure of the full arrangement. For instance, any region of a subarrangement can be further subdivided by the hyperplanes that were excluded from the subarrangement to form regions of the full arrangement. This hierarchical relationship is fundamental to many algorithms that analyze arrangements, as it allows for breaking down complex problems into smaller, more manageable parts. Studying subarrangements is crucial for understanding properties of arrangements, such as their incidence structure, connectivity, and the number of regions they create. They play a role in problems related to linear programming, configuration spaces, and polytope theory.