Suppose I have a random $n$-dimensional linear subspace of $\mathbb{R}^m$. How many of the $2^m$ hyperquadrants does this space intersect, on average? Alternatively, what are the odds that this subspace intersects the strictly positive hyperquadrant? What follows is the progress we've made so far:
Trivially, we have that the number of hyperquadrants $N(n,m)$ is $N(0,m)=0, N(m,m)=2^m$.
I think we can also show that each hyperquadrant either contains an element of the subspace or an element of its complement but not both. "but not both" follows from the fact that two vectors lying strictly in a hyperquadrant should have positive inner product. The claim that at least one lies in the hyperquadrant follows from the fact that if the claim failed one could form a basis for the entire $m$-dimensional space via vectors at the corner of the hyperquadrant which would not overlap with the equally spanning basis for the same space given by the subspace and its complement.
Given this, we have that each hyperquadrant can be assigned to either the subspace or its complement, and that therefore $N(n,m) + N(m-n,m) = 2^m$. This also implies that since $N(1,m) = 2, N(m-1,m) = 2^m-2$.
But how to make progress from here? I'm hoping there is some simple induction step that, for example, relates $N(n+1,m+1)$ to $N(n,m)$. But I can't find it. Any ideas?