$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$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
- Do you know choices for $n$ and $I$, so that $I_n$ has some 'geometric meaning'?
- What are some restrictions of the invariants that can be created in this way?
- Do you have a reference where this construction is carried out in detail?
Possible choices for $I$ may be $$ \diam(X)=\max_{x,y}d(x,y),\quad \rad(X)=\min_x \max_y d(x,y), \quad \iso(X)=\max_x \min_{y\neq x} d(x,y) $$ 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.
Example of above construction for explicit choice of $M$ and $I$
