Finding size of subspace I have a finite metric space $M$ with a known number of points, and a subspace of $M$ we will call $N$. I know the average value of $d(x,N)$ for $x\in M$, where $d(x,N)=\text{inf}_{y\in N} d(x,y)$. Using this information is it possible to measure (or even estimate) the size of $N$? 
What properties of $M$ and $N$ would I have to know in order to? 
This problem came up in a machine learning application, and is way out of my area but seems somewhat promising. Intuitively, the longer the mean length from $M$ to $N$, the smaller a proportion $N$ is of $M$, but I can't quantify this relationship.

Some thoughts.
Absent information about the distribution of distances among the pairs of points of $M$ I don't think there is much you can conclude. 
Suppose $M$ consisted of very many points distributed uniformly at random over the unit square (say). Then for any moderately large uniformly distributed subset $N$ the average value of $d(x,N)$ would be about the same, independent of the size of $N$.
For small closely packed clusters $N$ the average value of $d(x,N)$ would increase with the distance of $N$ from the center of the square, again independent of the size of $N$.
Even knowing more about $M$ I think the same kind of arguments may apply.  You can use subsets $N$ with the same approximate distribution of distances for the first examples and small clusters for the second.
