The analogy between these two quantities goes back to Steiner (Ueber parallele Flaechen, 1841). Both of them appear as coefficients of linear terms in the expansion of the area of the surface at distance $\epsilon$ from the given smooth/polyhedral surface.

Minkowski proved the convergence of the discrete to smooth for convex surfaces with respect to the Hausdorff metric on the space of convex bodies. Keywords: intrinsic volumes, quermassintegrals. This also implies that the total mean curvature is defined for arbitrary convex surfaces.

Besides, it can be defined for the boundaries of finite unions of convex bodies by sort of inclusion-exclusion formula.

The analogy between these two quantities goes back to Steiner (Über parallele Flächen, 1840). Both of them appear as coefficients of linear terms in the expansion of the area of the surface at distance $\epsilon$ from the given smooth/polyhedral surface.

Minkowski proved the convergence of the discrete to smooth for convex surfaces with respect to the Hausdorff metric on the space of convex bodies. Keywords: intrinsic volumes, quermassintegrals. This also implies that the total mean curvature is defined for arbitrary convex surfaces.

Besides, it can be defined for the boundaries of finite unions of convex bodies by sort of inclusion-exclusion formula.