An Identity for the Coefficients of Characteristic Polynomials of Hyperplane Arrangements

Abstract Consider a finite collection of affine hyperplanes in $$\mathbb R^d$$ R d . The hyperplanes dissect $$\mathbb R^d$$ R d into finitely many polyhedral chambers. For a point $$x\in \mathbb R^d$$ x ∈ R d and a chamber P the metric projection of x onto P is the unique point $$y\in P$$ y ∈ 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 $$\text {dim}(x,P)$$ dim ( x , P ) . We prove that for every given $$k\in \{0,\ldots , d\}$$ k ∈ { 0 , … , d } , the number of chambers P for which $$\text {dim}(x,P) = k$$ 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)].