Mean Width of Disjoint Union $\newcommand{\vol}{\operatorname{vol}}$Let $K$ be a convex body in $\Bbb R^n$ (a convex body is a convex, compact subset of $\Bbb R^n$ with nonempty interior). The mean width of $K$ is defined as $b(K)=\dfrac{2}{\vol_{n-1}(\partial B_2^n)} \int_{S^{n-1}} h_K(u) \,du$, where $h_K(u)=\sup_{x\in K}\langle x,u\rangle$ is the support function of $K$ and $du$ is shorthand for $dH^{n-1}(u)$. Is there a closed-form expression for the mean width of a disjoint union of convex bodies? Or perhaps a closed form expression for the support function of a disjoint union?
For example, for convex bodies $K$ and $L$ the volume (and surface area) satisfy a valuation property $\vol_n(K\cup L)=\vol_n(K)+\vol_n(L)-\vol_n(K\cap L)$ (and similarly for surface area), even when $K\cup L$ is not convex. Is this true for the mean width as well, i.e. is it true that $b(K\cup L)+b(K\cap L)=b(K)+b(L)$? If not, does one of the directions in the inequality always hold? Or, by what formula or inequality are the support function $h_{K\sqcup L}$ of a disjoint union and $h_K,h_L$ related?
My attempt at a solution: By Groemer's extension theorem, the mean width can be extended uniquely to a valuation on the class of polyconvex sets, so it satisfies the valuation property $b(K\cup L)+b(K\cap L)=b(K)+b(L)$. (seems fishy)
Thank you in advance for your help.
 A: these are actually two questions.


*

*If you define the "mean width" for non-convex bodies by the same
integral over $h_K(x)=\sup_{v\in K} \langle x,v\rangle$, the "mean width" of the union of two convex bodies will in
general depend on the concrete shape and the relative position of
the bodies.  Consider a simple example:
Let $K$ be a unit disk centered at $(-d/2,0)$ and  $L$ be a unit
disk centered at $(d/2,0)$, $d>2$ Then $b(K)=b(L)=2$ and $b(K\cap
L)=b(\emptyset)=0$. However, if an element of $S^1$ is represented
by $u\in [0,2\pi]$:
\begin{align*} b(K\cup L)&=\frac{1}{\pi}\int_0^{2\pi} (1+ d/2 |\cos
u|) du \\   &=  2 +  \frac{2d}{\pi} \int_0^{\pi/2} \cos u du = 2+
\frac{2d}{\pi} \; . \end{align*}
Hence there can't be a formula expressing this integral just in
terms of $b(K)$, $b(L)$ and $b(K\cap L)$.

*The second question is, whether $b$ can be extended from the class
of convex bodies to the class of arbitrary unions of convex bodies,
such that additivity holds. The answer is yes. The mean width is a valuation for convex bodies, as the support function mapping is a valuation, s. e.g. Rolf Schneider: Convex Bodies: The Brunn–Minkowski Theory, p. 330. See also p. 331 for an explicit additive extension of the support function and thus the mean width. (Groemer's extension theorem might have been used instead to prove the existence of such an extension.)
The extension of the support function looks like $$h_K(x) = \sum_{\lambda \in {\mathbb R}} (\chi(H_{x,\lambda}\cap K) - \lim_{\mu\searrow \lambda} \chi(H_{x,\mu}\cap K)) \; ,$$ and the integral definition of the mean width with this extension of the support function yields the extension of the mean width.
