Proving the "centroid" property and the existence of corresponding convex polyhedron in Minkowski Problem

Question

Assume $P$ is a convex polyhedron embedded in $\mathbb{R}^{3}$, the faces are $\left\{F_{1}, F_{2}, \cdots, F_{k}\right\}$, the unit normal vector to the face $F_{i}$ is $\mathbf{n}_{i}$, the area of $F_{i}$ is $A_{i}, 1 \leq i \leq k$.

$~~~~~~~~~~~~~~~~~~~~~~~~$

• Show that $$ A_{1} \mathbf{n}_{1}+A_{2} \mathbf{n}_{2}+\cdots A_{k} \mathbf{n}_{k}=\mathbf{0} $$
• Given $k$ unit vectors $\left\{\mathbf{n}_{1}, \mathbf{n}_{2}, \cdots, \mathbf{n}_{k}\right\}$ which can not be contained in any half space, and $k$ real positive numbers $\left\{A_{1}, A_{2}, \cdots, A_{k}\right\}, A_{i}>0$, and satisfying the above condition, show that there exists a convex polyhedron $P$, whose face normals are $\mathbf{n}_{i}$ 's, face areas are $A_{i}$ 's.$$$$ In my opinion the easy case of this problem is when it's the $\mathbb{R}^{2}$ case, the sum of area of the triangle times the corresponding unit vector is $0$ by properties of the centroid, but I have no idea of how to extend this property in to 3-dimension. Is there any hint for continuing here?

Answer 1

The identity

$ \displaystyle \sum_{i=1}^k A_i \mathbf{n}_i = 0 $

can be proven using the divergence theorem which states that the surface integral of a vector field $\mathbf{F}(x,y,z)$ over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence of $\mathbf{F}$ over the region inside the surface.

This is expressed in mathematical terms as follows

$ \displaystyle \int_D \nabla \cdot \mathbf{F} dV = \int_S \mathbf{F} \cdot \mathbf{n} dS $

Now if we select $\mathbf{F} = \mathbf{c}$ , a constant vector, then the divergence of this constant vector field is $0$, hence

$ 0 = \displaystyle \int_S \mathbf{c} \cdot \mathbf{n} \ dS = \mathbf{c} \cdot \int_S \mathbf{n} \ dS $

Since we can choose $\mathbf{c}$ to point in any direction, it follows that it must be the case that

$ \displaystyle \int_S \mathbf{n} \ d S = \mathbf{0} $

Finally, in our case this last surface integral is just the summation

$\displaystyle \sum_{i=1}^k A_i \mathbf{n}_i $

Thus we have shown that it is the zero vector for any polyhedron.

This answers the first part of your question. I wasn't able to come up with anything concerning the second part.

Answer 2

The first identity is a consequence of the fact that volume is translation invariant. For each each face $F_i$, let $h_i$ be $n_i\cdot x$ for any $x \in F_i$. The volume of the polyhedron is $$ \frac{1}{3}(h_1A_1+\cdots + h_kA_k). $$ If you translate the body by a vector $v$, then each $h_i$ becomes $$ \tilde{h}_i = (x+v)\cdot n_i = h_i + v\cdot n_i, $$ which implies that the volume is also given by \begin{align*} V &= \frac{1}{3}(\tilde{h}_1A_1 + \cdots + \tilde{h}_kA_k)\\ &= \frac{1}{3}(h_1A_1+ \cdots + h_kA_k) + \frac{1}{3}v\cdot(n_1A_1 + \cdots + n_kA_k)\\ &= V +\frac{1}{3}v\cdot(n_1A_1 + \cdots + n_kA_k). \end{align*} Therefore, $$ \frac{1}{3}v\cdot(n_1A_1 + \cdots + n_kA_k) = 0. $$ Since this holds for any direction $v$, it follows that $$ n_1A_1 + \cdots + n_kA_k. $$

The second question is known as the Minkowski problem, which was originally posed and solved by Minkowski. I'm not sure where you can find a proof specifically for polyhedra. You can try looking in Schneider's book, Convex Bodies: the Brunn-Minkowski Theory