Reading Ravi Vakils Monthly article of february 2011 and watching the video; he mentions that the Dehn invariant is related to the linear invariant measure $\mu_1$ of geometric probability. The Klain Rota book discusses the Dehn invariant on pages 115-116. To make the question self contained:
Let $P \subset \mathbf{R}^3$ be a convex polyhedron. Let $rP$ be the dilation of $P$ by the positive real number $r$.
The Dehn invariant of $P$ is an element of the real vector space $\mathbf{R} \otimes_{\mathbf{Z}} \mathbf{R}/(\pi\mathbf{Z})$, given by the formula $$ D(P) = \sum_e \mu_1(e) \otimes (\text{angle between the two faces containing }e) $$
The mean width $W(P)$ is a real scalar, given by generally by total integrated mean curvature over the surface but specially: $$ W(P) = \sum_e\mu_1(e)(\text{angle deficiency from being flat of }e) $$
Both $D$ and $W$ scale linearly with dilations: $D(rP) = rD(P)$ and $W(rP) = rW(P)$. And the Dehn invariant is invariant under the scissors congruence relation, while the mean width doesn't seem to.
Is this it or am I missing something?
