Define the caliper diameter of a polyhedron as follows:

Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the smallest possible distance allowing the whole of the polyhedron to lie in the region of space between the two planes. Define the perpendicular distance between the two planes as the caliper diameter of the polyhedron.

How do I prove that the average caliper diameter of the polyhedron across all possible rotations is given by this formula:

$$\sum_{e\in E} L_e(\pi - \delta_e)/(4\pi)$$

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 $\pi/2$).