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