Here are some definitions (in dimention 3, but you can easily generalize):
Definition : Given a finite number of points with coordinates $ P_1 = (x_1,y_1,z_1), .., P_n = (x_n,y_n,z_n) $, a convex solid is the convex hull of these points, i.e. all the points of the space defined as $\sum_{1\leq i \leq n} \lambda_i P_i $ with $\forall i \in [1,n], \lambda_i \in \mathbb{R}, 0 \leq \lambda_i \leq 1, \sum_{1 \leq i \leq n } \lambda_i = 1 $
For instance, the points (0,0,0); (1,0,0); (0,1,0); (1,1,0); (0,0,1); (1,0,1); (0,1,1); (1,1,1) define a cube.
Rq : If I add the point (0.5,0.5,0.5), I define the same cube.
Definition : An oriented plane is 4 real numbers $a, b, c$ and $d$.
Definition : An oriented plane is a face of a solid, iff at least three points defining the solid verify $ax+by+cz = d$ (are in the plane), and all points of the solid verify $ax+by+cz \leq d$.
For instance, the plane defined by $a=1, b=0, c=0, d=1$ in the cube described above is a face. The one defined $a=1, b=1, c=0, d=1$ is not, since it fails the second condition of the definition. The one defined by $a=-1, b=0, c=0, d=-1$ is not either. Indeed, the definition specifies $ax+by+cz \leq d$, and it is not the case here.
The definition of a face given here enables to control on which "side" of the plane the solid is. Putting a "-" sign in front of all the coefficients defining a face would mean the solid is on the other side of the plane.
Definition : A view direction is a vector $ (x_v,y_v,z_v) $ with $x_v^2+y_v^2+z_v^2=1$.
The direction can be identified at the way you look at. Here, it is assumed you look from a point at an infinite distance.
Definition : A face $(a,b,c,d)$ of a solid is visible from a direction $ (x_v,y_v,z_v) $ iff $ax_v+by_v+cz_v < 0 $.
Here is the proof (short version) for the cube (with the points described earlier).
All faces of this cube are defined by the following quadruples (and no others) :
(1,0,0,1)
(0,1,0,1)
(0,0,1,1)
(-1,0,0,0)
(0,-1,0,0)
(0,0,-1,0)
So, with any viewing direction, one sees at most three faces (the ones corresponding to the sign of your viewing direction).
(you see less if one of the viewing direction coordinates is 0, it would mean we see only some of its edges).
For a different kind of solid, you need its defining points, and then the proof would basically go this way
first find all faces of this solid
assume there is another face and discover it is one you already have (so you have proven that you have all of them)
Assume you have a viewing direction. Then, a lot of cases to look at, like "if my direction is such that $x_v + ... < ...$"
find, in all the cases, the maximal number of faces
If you think there is a leaner/more conventional way to define the problem, do not hesitate to mention it.