The mean caliper formula was derived in 1967 by the metallurgist John Cahn [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. behind a paywall
[2] R.E. Miles, Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats, Adv. App. Prob. 1 (1969) 211-237.
[3] R.E. Miles, Direct Derivations of Certain Surface Integral Formulae for the Mean Projections of a Convex Set, Adv. Applied Prob. 7 (1975), 818-829.