Reachability of points on Manifolds I was thinking with a friend that on a surface some points are more reachable than other.In the sense that their average distance to the other points is lower.
e.g. suppose that we have a circle in a 2D space, then its central points will be more "reachable" than its perimeter points. While on a sphere all its point have the same "reachability".
$$C=\left\{p=(x,y) \in \mathbb{R}^2\;|\;x^2+y^2 \leq R \right\}$$
Let $ p_0 \in C$
$$\frac{1}{|C|} \int_{C} ||p-p_0||_2 dp = \frac{1}{|C|} \iint_{C} \sqrt{(x-x_0)^2+(y-y_0)^2}dx dy$$
There is some theory that talks about this concept?

"Laissons les jolies femmes aux hommes sans immagination" Marcel Proust
 A: It's a notion that's gotten some attention in computer graphics and vision over the last decade or so. Here's why: suppose you define the reachability of $P$ as the average over all $Q \in M$ of $d(P, Q)$ (or some power of the distance, like, say, the squared distance). Then this gives you a real-valued function $r$ on the manifold $M$, and if you're lucky, it'll actually be a Morse function. The morse-theory structure of $M$ under this function tells you something interesting about shape, and has some invariants under not just isometry, but other deformations as well.
For instance, if $M$ is the surface of my body, then my fingertips are likely to be extrema of $r$, i.e., critical points of index $0$ (or $2$, depending on whether I've got the sign right or not...). That'll be true even if I wave my arms around, or wiggle my fingers, or walk, or run...so under the 'near isometries' of typical human motion, these critical points serve as a kind of marker for the topological structure of the body. 
What's a pity is that I can't give you any references for this except to say that the first papers I saw about it were probably in SIGGRAPH, sometime in the early 2000s. 
