Let $S^d = \{x \in \mathbb{R}^{d+1}: \|x\|_2 = 1\}$ be the unit sphere in $\mathbb{R}^{d+1}$.
Given $n \in \mathbb{N}$, I want to understand how far apart each point from $S^d$ will be to its closest neighbor and how small the maximum of these distances may become across all points. I am trying to find:
$$\min_{\{x_1, …, x_n\} \in S^d} \max_{i \in [n]} \min_{j \neq i} \|x_i – x_j\|$$
This will depend on $n,d$ of course and I guess it will be optimal to space the points equally across the sphere.
I am mostly interested in asymptotic bounds on this expression.
In two dimensions it should be easy to get them because we can simply cut an arc of length $2\pi$ into $n$ equal parts.
Best Answer
This is not a full answer and if anyone knows more about this topic, please feel free to add further information.
Having followed some of the links suggested in the comments, I at least got the asymptotic bound I was mostly interested in.
It turns out that for every $d \in \mathbb{N}$ there exists a constant $K>0$ and a sequence of partitions $\mathcal{P}_N$ of the unit sphere $S^d \in \mathbb{R}^{d+1}$ into $N$ regions of equal area such that for all $N \in \mathbb{N}$ we have $$ \sup_{P \in \mathcal{P}_N} \; diam(P) \le K \cdot N^{-1/d}$$ where the diameter of a region $P$ is defined as $$diam(P) = \sup_{x,y \in P} \|x-y\|_2$$ The existence of such a partition was asserted by Stolarsky and a construction was given by Feige and Schechtman.