Minimum number of points for maximum distance in n dimensions

Question

Given an n-dimensional unit cube ([0, 1]^n), how would I determine the minimum number of points needed, so that I could distribute the points inside the cube and the distance between any other (not chosen) point inside the cube and the closest placed point is at most a given distance d?
And how would one place these points?

Answer 1

It seems you are asking for covering a cube by congruent spheres. Let me try to translate down to $\mathbb{R}^2$.

Let $S$ be a unit square in the plane. Define $r_\min(k)$ as the minimum radius $r$ such that $k$ disks of radius $r$ can be placed to cover $S$ completely. Under such a placement,

the distance between any other (not chosen) point inside the cube and the closest placed point is at most a given distance $r$

because it falls within a disk of radius $r$ (where I have changed the OP's $d$ to $r$, for radius).

So, for example, an optimal placement of $7$ points in a square is as shown below:

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  ^{Nurmela, Kari J., and Patric RJ Östergård. "Covering a square with up to 30 equal circles." (2000).
  CiteSeer link.}

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There is a complex literature on these problems, which have considerable practical significance, e.g.,

Huang, Chi-Fu, Yu-Chee Tseng, and Li-Chu Lo. "The coverage problem in three-dimensional wireless sensor networks." Global Telecommunications Conference, 2004. GLOBECOM'04. IEEE. Vol. 5. IEEE, 2004.