The surface area measure in terms of support functions $\def\RR{\mathbb{R}}$Let $K$ be a closed bounded convex body in $\RR^n$. The support function $h_K$ on $\RR^n$ is defined by
$$h_K(v) = \max_{w \in K} \langle v,w \rangle.$$
Let $S^{n-1}$ be the unit sphere. The surface area measure is the measure on $\sigma$ such that, for an open set $U$ in $S^{n-1}$, the measure $\sigma(U)$ is the $(n-1)$-dimensional Lebesgue measure of the set of $w \in \partial K$ where at which supporting hyperplanes normal to $v$ make contact. (I believe this is fairly standard terminology; I am reading Schneider's "Convex Bodies
The Brunn-Minkowski Theory" as my reference.)
Now, if $K$ is smooth, then $\sigma$ is a smooth $(n-1)$-form, and we can compute $\tfrac{\sigma}{\mathrm{Area}}$ in terms of the Hessian of $h$. Namely, restrict $h$ to the affine hyperplane $v+v^{\perp}$. (In other words, the tangent plane to $S^{n-1}$ at $v$.) Then $\tfrac{\sigma}{\mathrm{Area}}$ is the determinant of the Hessian of this restricted function.
On the other hand, suppose that $K$ is a polytope, so $h$ is piecewise linear. Then $\sigma$ is an atomic measure, concentrated on the normals to the facets of $K$. Then we can also compute $\sigma$ in terms of $h$ restricted to $v+v^{\perp}$: For $u$ in $v^{\perp}$, take the directional derivative $\tilde{h}(u) := \lim_{t \to 0^+} \tfrac{h(v+tu)-h(v)}{t}$. Then $\tilde{h}(u)$ is (I believe) the support function of the facet normal to $v$, and $\sigma(v)$ is the volume of that facet. We can recover the facet, and hence its volume, as the dual of $\tilde{h}$.
What I am trying to understand is how to interpolate between these formulas. In general, is $\sigma$ something in terms of the restriction of $h$ to $v+v^{\perp}$. And why am I seeing second derivatives in the smooth case and first (directional) derivatives in the polytopal case?
 A: I came to convex geometry from differential geometry and had many similar questions, too.
One way to view surface area measure is the following: The normal to a supporting hyperplane is essentially the Gauss map from $\partial K$ to $S^{n-1}$. This is a multi-valued map that is differentiable whenever it is single-valued. The points where it is multi-valued has measure zero on $\partial K$. The surface area measure is the measure $dA$ on $\partial K$ pushed forward to $S^{n-1}$ by the Gauss map.
At a point on $\partial K$, where the Gauss map has invertible Jacobian (i.e., $\partial K$ has positive Gauss curvature), the change of measure is given in terms of the the determinant of the Jacobian, which is Gauss curvature. In particular, Gauss curvature $\kappa$ satisfies $du = \kappa\,dA$. Therefore, if you pull back $dA$ by the inverse Gauss map (which is well-defined here), you get $dA = f\,du$, where $f$ is what convex geometers call the curvature function and is the reciprocal of the Gauss curvature as a function of the unit normal.
On the other hand, the Gauss map is constant along a flat face, So if you push forward $dA$, you get an atomic measure at the normal to the face with mass equal to the area of the face.
As for how all this is connected to the support function, the observation is that, wherever $h$ is differentiable, its gradient is the inverse Gauss map.  So the Jacobian of the inverse Gauss map is the Hessian of $h$. It's also worth noting that the support function of the polar body is the reciprocal of what is called the radial function. And the gradient of the polar support function, if normalized appropriately, is really the Gauss map itself.
What's cool about all of this is that, even though the above mentions the unit sphere and the Gauss map, all of these concepts do not require the Euclidean structure of $\mathbb{R}^n$. The only structures needed are the vector space structure and Lebesgue measure. A lot of what's in Schneider's book is invariant or equivariant under the action of $SL(n)$. But, at least for me, this is hard to see using the standard approach. So I worked out my own way of doing things. The first thing I always do is view the body $K$ as a subset of an abstract vector space $V$, the polar body is then in $V^*$, and the domain of the support function is $V^*$. Figuring out how to describe surface area measure in this setting took a bit more work.
I developed my own formulation of all of this, avoiding any use of the dot product or the unit sphere, and wrote it up in a survey paper, Affine Integral Geometry from a Differentiable Viewpoint.
