For some polyhedron, $P$, define the mean width function,
$$H(P)=\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$
Where $E$ is the set of all edges of the polyhedron, $L_e$ is the length of edge $e$ and $\delta_e$ is the interior angle where the two faces forming edge $e$ meet (e.g. for a cube the interior angle between two faces is always $\pi/2$). (Note: this may not actually represent the mean width for a concave shape).
My question is, for possibly concave and possibly intersecting polyhedra, $P$ and $Q$, does the following 'valuation property' hold and, if so, where could I find a proof of this:
$H(P\cup Q) = H(P) + H(Q) - H(P \cap Q)$
I did find these questions (here and here) about the "valuation property" but the answers seem to disagree on whether this holds for concave polyhedra!
Also, I feel I should mention I have posted a similar question on Math Overflow here but in this question I was interested in the interpretation rather than the result.