Let $\vec{A}_1, \vec{A}_2, \dots, \vec{A}_n$ be the outward pointing normals to the surfaces of an n-valent polyhedron, whose magnitudes $A_l=|\vec{A}_{l}|$ are the areas of the faces. These normals correspond to a closed polyhedron:
\begin{equation}
\vec{A}_1 + \dots + \vec{A}_n = 0.
\end{equation}
According to equation 36 of https://arxiv.org/pdf/1208.2228.pdf we can express the volume of a 4-valent polyhedron in terms of these normals by:
\begin{equation}
V=\frac{\sqrt{2}}{3}\sqrt{|\vec{A}_{1}\cdot(\vec{A}_{2}\times\vec{A}_{3})|}.
\end{equation}
I would like to obtain an expression for the volume of a general n-valent polyhedron, such that it only depends on the normals to the surfaces, $\vec{A}_l$. I have been searching and all the expressions I have found require additional information, such as the equation given by (https://en.wikipedia.org/wiki/Polyhedron#Volume):
\begin{equation}
V=\frac{1}{3}\left|\sum_{j}(Q_j\cdot\mathbf{N}_j)\text{Area}(S_j)\right|,
\end{equation}
where $Q_j$ is any point on the surface $S_j$ and $\mathbf{N}_j$ is a $\underline{\text{unit}}$ outward point vector normal to $S_j$.
Summarizing, is there a way to obtain the volume of a polyhedron only from the normals to the faces, $\vec{A}_{l}$?
Best Answer
I don't think there is a closed form formula for what you want. But if you can reconstruct the polyhedron from the given data, you can find its volume.
If the polyhedron is convex, it is uniquely determined (up to translation) by the face normal data. This is generally referred to as the Minkowski Problem. (You can do searches on "Minkowski Problem polyhedron" to get more information. Here's an introductory slide presentation.)
A cursory search indicates that reconstruction methods are numerical. An early iterative method for polyhedra is given in Little, Recovering shape and determining attitude from extended gaussian images. (See also An Iterative Method for Reconstructing Convex Polyhedra from Extended Gaussian Images.)