Consider a metric space $(M,d)$ and consider a collection of points $X_n := \{x_1,\dots,x_n\} \subset M$. Let $$ N_\epsilon(y;X_n) := | \{ x \in X_n: d(x,y) \le \epsilon \}| $$ where the RHS is the cardinality of a set. For a set $Y \subset M$, define $$ N_\epsilon(Y;X_n) = \inf_{y \in Y} N_\epsilon(y;X_n) $$ Does this quantity look familiar, or is related to a more standard object?
Let us say that $M$ is the standard Euclidean space $\mathbb{R}^d$ and $X_n$ is a random sample from a distribution. What can be said about the asymptotic behavior of $ N_{\epsilon_n}(Y;X_n) $, as $n \to \infty$ and possibly $\epsilon_n \to 0$.