TITLE: Expected number of vertices of a hypercube slice -- is this new/interesting?
QUESTION [24 upvotes]: I am a (mostly) amateur mathematician, but my education and work have featured a lot of mathematics, and recently I bumped into a mathematical problem for which I can find no references, and I am looking for commentary on whether the problem has been looked at before and whether it is possibly of interest to working mathematicians.  
The problem is this: What is the expected number of vertices for a random slice of a (hyper-)cube?  The answer depends on what distribution of slices you use.  There are some distributions which seem more "natural" than others---e.g. since a slice is characterized by an orientation and a translation, it would be natural to expect that the orientational part had some rotation symmetry (at least the symmetry of the hypercube).  In particular, the most natural distributions to me are:

Choose the orientation uniformly, and then choose a translation uniformly from all translations for which the resulting slice still intersects the hypercube. 
Choose the orientation uniformly, and the choose a point uniformly in the hypercube for the slice to pass through.
Choose the orientation uniformly, and let the slice pass through the hypercube center. 

I can write down expressions for the expected number of vertices in each of these three cases.  The first is the most interesting---for a k-dimensional slice, the expected number of vertices is $2^k$, regardless of the dimension of the hypercube.  I found this surprising---for example, this says that a 2-slice will have 4 vertices on average, whereas any particular 2-dimensional slice of an n-dimensional hypercube can have anywhere from 3 to 2n vertices.  For the other two, the expressions involve integrals which I can evaluate in some, but not all cases.  The answer is not as elegant in either case, though.  
The proof for the first distribution involves a neat geometric observation, which is a formula for the volume of a projected hypercube (or projected parallelepipid, in general).  This is sort of like the determinant formula for the volume of a parallelepipid generated by a set of linearly independent vectors, except that here the vectors need not be linearly independent.  Perhaps such a formula is of interest in its own right?
Anyway, my questions are whether any of this is new or interesting.  I can fill in more details if that would help.

REPLY [2 votes]: Just to give you some positive feedback, I am interested on finding the largest possible hypercube where the slice is fully contained inside another convex set. 
In order to check that the slice is completely inside a convex set represented by $Ax \leq b $, it is useful to represent it in V-Rep, as the convex combination of its vertices. Then I can easily check for all vertices.
In order to know how bad can this get, it is useful to know how many vertices we have on average.