**A cheap way of defining invariants of Riemannian manifolds?**

Let $(M, g)$ be a Riemannian Manifold with finite volume and $X=(X_1,\dots, X_n)\sim \mathscr U(M^n)$ a random sample with respect to the probability measure induced by $g$. That is, we have $n\in \mathbb N$ iid uniform points in $M$, so that $X$ is a (random) finite metric subspace of $M$.

Now let $I$ be your favorite (real valued) invariant of finite metric spaces. Then $I(X)=I\circ X$ is a real random variable and $I_n=\mathbb E[I(X)]$ (when existent) is an invariant of the *Riemannian* manifold $M$.

**Questions**

 1. Do you know choices for $n$ and $I$, so that $I_n$ has some 'geometric meaning'?
 2. What are some restrictions of the invariants that can be created in this way?
 3. Do you have a reference where this construction is carried out in detail?

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Possible choices for $I$ may be
$$
diam(X)=\max_{x,y}d(x,y),\quad rad(X)=\min_x \max_x d(x,y), \quad iso(X)=\max_x \min_{y\neq x} d(x,y)
$$
or with $\varepsilon >0$
$$
\underline c _\varepsilon(X) = \min_x\#\{y\vert d(x,y) < \varepsilon\},\quad \bar c _\varepsilon(X) = \max_x\#\{y\vert d(x,y) < \varepsilon\},
$$
$$
com_\varepsilon(X)=\#\{\text{components of }x\sim y \iff d(x,y)<\varepsilon\},\quad con(X)=\inf\{\varepsilon >0\,\vert\, com_\varepsilon(X)=1\}
$$
and many more.

----

[Same question on MS][1]

[Example of above construction for explicit choice of $M$ and $I$][2]


  [1]: https://math.stackexchange.com/questions/4855023/statistical-invariants-of-riemannian-manifolds
  [2]: https://math.stackexchange.com/questions/4829402/when-does-a-random-geometric-graph-become-connected