Let $(\mathcal{X}, d)$ be a metric space and $n \in \mathbb{N}$. Find
$$\operatorname*{argmin}_{\mathcal{Y} \in [\mathcal{X}]^n} \sup_{x \in \mathcal{X}} \inf_{y \in \mathcal{Y}} d(x, y)$$
That is, find a set of $n$ points that minimizes the maximum distance between any point and the closest point in the set.
I am especially interested in the case where $d$ is the Euclidean metric and $\mathcal{X}$ is any of the following:
- unit L$_p$ ball: $\|x\|_p \leq 1$
- unit L$_p$ sphere: $\|x\|_p = 1$
- unit simplex: $x \geq 0, x \cdot 1 = 1$
for any of $p \in \{1, 2, \infty\}$. Does this problem have a name in the literature? Are there any efficient algorithms for obtaining exact or approximate solutions in these special cases?