TITLE: Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k? 
QUESTION [6 upvotes]: Consider $S^k \subset  R^{k+1} $.  Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ).
Consider Voronoi cell around each point. 
How many neighbours would a cell have ? I mean neigbours are the cells which have non empty intersection. "How many" means average over distribution. (Clearly it is less than N,
but what is it behaviour ? N/C, sqrt(N) or what ?)
Actually I more interested not about the sphere but about the cube: take unit cube $[-1, 1]^k$.
And take randomly some number $N$ of its vertexes.
The same questions.

Motivatation:
As I tried to explain in this MO quest these problems are related to decoding noise signal. This question can be translated in this language as follows - if there chance to do some "preprocessing" such that it would significantly reduce decoding complexity.
I mean in the answer is much smaller than N, then yes, otherwise, not.

REPLY [4 votes]: This was studied by Miles in the seventies. A good reference is Schneider and Weil's book. The relevant results are in section 10.2