The mean caliper formula was derived in 1967 by the metallurgist <A HREF="https://en.wikipedia.org/wiki/John_W._Cahn">John Cahn</A> [1]. Cahn allowed for a curvature $K$ of the faces and assumed that all dihedral angles $\pi-\delta_e$ are the same, equal to $\epsilon$. Then the average curvature $\langle K\rangle$ is related to the average caliper $\langle C\rangle$ by
$$\langle K\rangle=\frac{2\pi}{A}\left(\langle C\rangle-\frac{\epsilon}{4\pi}\sum_e L_e\right)$$
where $A$ is the total area of the faces and $L_e$ is the length of edge number $e$.

A derivation for $\langle K\rangle=0$ that also allows for varying $\delta_e$ is given in [2] (page 231) and in [3] (page 825).

[1] J.W. Cahn, *The significance of average mean curvature and its determination by quantitative metallography*, Trans. Met. Soc. AIME 239 (1967) 610-616.
<sub>behind a <A HREF="http://www.aimehq.org/resources/library">paywall</A></sub>    

[2] R.E. Miles, <A HREF="https://www.jstor.org/stable/1426218?seq=1#page_scan_tab_contents">Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats,</A> Adv. App. Prob. 1 (1969) 211-237.

[3] R.E. Miles, <A HREF="https://www.jstor.org/stable/1426401?seq=1#page_scan_tab_contents">Direct Derivations of Certain Surface Integral Formulae for the Mean Projections of a Convex Set</A>, Adv. Applied Prob.
7 (1975), 818-829.