$\newcommand{\Unif}{\mathop{\rm Unif}} \newcommand{\argmin}{\mathop{\rm argmin}}$ Let A be a set in the plane (we could have something more general here). Let $X_k\sim \Unif(A)$, $k=1,2,\dots, n$. Denote $X=\{x_1,x_2, \dots, x_n\}$.
Define
$$g(x) = \argmin_{y\in X\setminus \{x\}} |x-y|$$
and then the random variable
$$M=\#X\setminus g(X).$$
I'm interested in how the set $A$ affects $\mathbb{E}[M]$.
Notes:
- The parameter $n$ is at loose here, but we can consider it going to infinity.
Also, consider the set $A$ to be open and bounded (the discrete case ($A$ being finite, for the uniform distribution to exist) might also be interesting but I'd like to think of the "shape" of the set).
$A$ could also be a sphere or torus, if you like.