Consider projections of points in Rⁿ onto chambers of a hyperplane arrangement. Each coefficient of the characteristic polynomial is proportional to the average spherical volume of the set of points that project onto faces of a fixed dimension. As a corollary, for a finite reflection arrangement each coefficient gives the precise spherical volume of the points that project onto the faces of a fixed dimension of the fundamental chamber.

This talk reflects joint work with Mathias Drton and Ed Swartz.