An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements

Consider a finite collection of affine hyperplanes in R-d. The hyperplanes dissect Rd into finitely many polyhedral chambers. For a point x is an element of R-d and a chamber P themetric projection of x onto P is the unique point y is an element of P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by dim(x, P). We prove that for every given k is an element of{0,..., d}, the number of chambers P for which dim(x, P) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873-2887 (2010)].