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 weird real vector space $\mathbf{R} \otimes\_{\mathbf{Z}} \mathbf{R}/(\pi\mathbf{Z})$, given by the formula
$$
D(P) = \sum\_e (\text{length of }e) \otimes (\text{angle between the two faces containing }e)
$$


The mean width $W(P)$ is a real scalar, given by the formula
$$
W(P) = \int\_{\vec{u} \in S^2} \left(\text{length of the interval }\left\{\vec{v} \cdot \vec{u} \mid \vec{v} \in P\right\}\right)
$$


Both $D$ and $W$ scale linearly with dilations: $D(rP) = rD(P)$ and $W(rP) = W(P)$. And they are both invariant under the scissors congruence relation.


Is there a formula for $W$ in terms of $D$?



Is mean width a Dehn invariant?
Additivity in the sense of scissors congruence means that if $A$ and $B$ have disjoint interiors, then $$D(A \cup B) = D(A) + D(B).$$ 


Valuations such as the mean width satisfy $$W(A \cup B) = W(A) + W(B) - W(A \cap B).$$


That last term is not necessarily $0$ when $A$ and $B$ have disjoint interiors. For example, in two dimensions, let $A$ and $B$ be (closures of) rectangular halves of a unit square. $A\cap B$ is a unit line segment. In two dimensions, the mean width is the perimeter of the convex hull divided by $\pi$. $W(A) = W(B) = 3/\pi$. $W(A \cup B) = 4/\pi$. $W(A\cap B) = 2/\pi$. So, the mean width is not additive in the sense of scissors congruence, as was pointed out in the comments. The last term is necessary in $4/\pi = 3/\pi + 3/\pi - 2/\pi$.


In three dimensions, the mean width is not determined by the Dehn invariant together with volume, since there are figures with equal Dehn invariants and volumes but different mean widths, such as the rectangular solids given by Ilya Bogdanov. 


