the induction in the proof of our main theorem. In Appendix A of their article [@GKM], Goresky–Kottwitz–MacPherson show that a certain function, which they call $\psi_C(x,\lambda)$, is a *valuation* (see Definition \[def\_valuation\]) on closed convex polyhedral cones, although they do not phrase it in these terms. We show that their function is a special case of a general construction that takes two valuations and produces a third one via a signed convolution. See Definition \[definition\_signed\_convolution\] for the precise definition of this signed convolution.

In Appendix \[appendix\_2\_structures\] we review the theory of $2$-structures, due to Herb; see for example Herb’s review article [@Herb-2S]. We believe that this will be useful to the reader for a number of reasons. The proofs of the fundamental results of this theory are somewhat scattered in the literature and sometimes left as exercises. Furthermore, we needed to slightly adapt a number of results so that they continue to hold for Coxeter systems that do not necessarily arise from a (crystallographic) root system.

Hyperplane arrangements {#section_hyperplane_arrangements}
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Background material {#background}
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We fix a finite-dimensional ${{\mathbb{R}}}$-vector space $V$ with an inner product $(\cdot,\cdot)$. If $\alpha\in V$, we write $$\begin{aligned}
H_\alpha & = \{x\in V : (\alpha,x)=0\}, &
