A Note on Hyper-Plane Arrangements in R^d

This note addresses hyper-plane arrangements in R and functions that are constant in the interior of each of the d-dimensional faces of the arrangement. We show that such a function g can be expressed in a simple form using basis functions that are products of d or less indicator functions of the open half-spaces bounded by the hyper-planes in the arrangement. Moreover, we present a simple and efficient algorithm that can be used to express g as a linear combination of these basis functions.