Maximum average Manhattan distance to nearest neighbor

Question

When you place points on a plane and measure the Manhattan distance for each point to its closest other point, what is the maximum average distance you can get for $n$ points?

For example, if we have $2$ points on a $15 * 15$ plane, we can place both points in opposite corners and have a distance of $15 + 15$ for both points to the other, thus an average distance of $15$. This is the maximum value we can get, and hence it is the maximum average distance for $2$ points.

The cases where $2 \leq n \leq 5$ are visualized here. It seems that for the first few rounds, we simply divide the circumference by the number of points (i.e. the maximum average distance is $\frac{60}{n}$). For $n = 5$ however, we gain a free place to put a point.

My questions:

1. What is the formula behind this?
2. How does this translate to higher dimensions?
3. How do we continue placing points such that the average distance is maximum?

Answer 1

I believe your question is related to packing, e.g.,

van Dam, Edwin R. "Two-dimensional minimax Latin hypercube designs." Discrete Applied Mathematics 156, no. 18 (2008): 3483-3493. Journal link.

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