Abstract
AbstractConsider 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)].
Funder
Westfälische Wilhelms-Universität Münster
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Theoretical Computer Science
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