(Calculus of) Variations on a Vesper Theme

Calculus Level 5

$$P_{1}, P_{2}, \ldots, P_{n}$$ are points on the surface of the unit sphere. Define $$D_{n}$$ as the set of all possible distances between any two of these points.

Find

$\sum_{n=2}^{6} \min_{d \in D_{n, \sigma}} d$

where $$\sigma$$ is some distribution of $$P_{1}, P_{2}, \ldots, P_{n}$$ such that the mean distance between points is maximised.

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