# (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.