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How can I generate a set of $N$ points that minimize $\sum_{x \in S^3} \min_{n \in N} \|x-n\| $, the distance between all points in the unit sphere and the closest point in the set? Intuitively, this should be some sort of grid on the sphere, but for some reason I haven't been able to formalize this, or even successfully Google it. A solution, or even just the proper terminology for such a set, would be greatly appreciated.

Generalizations to $S^d$ and a method for sampling these sets would be great :)

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  • $\begingroup$ I'm confused if $x \in S^3$, how can you talk about summation (which needs countability)? the definition you might want to use is $$dist(n,S^3) = \inf_{x \in S^3} \left\lVert n - x \right\rVert$$ $\endgroup$ – user8469759 Mar 15 '18 at 17:53
  • $\begingroup$ I might an idea, assuming this is what you want. $\endgroup$ – user8469759 Mar 15 '18 at 18:22
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Maybe you want to parameterize based on $$I_N = \int_{\Sigma = S^3} d(x,N) d\Sigma(x).$$ Because you have $N \subset \mathbb{R}^3$ and $|N|$, you have a cost function $f(y) : \mathbb{R}^{3|N|} \to \mathbb{R}$ defined as $$f(y) = \int_{\Sigma = S^3} d(x,N) d\Sigma(x).$$

The vector $y$ encodes, sorted, all the coordinates in the set $N$, and by definition you have a lot of symmetries, for example if $P$ is a permutation matrix (permuting points though, not single coordinates) it is easy to prove that $f(Py) = f(y)$.

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