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Generaliation the result in our paper for sum and similarly my previous question for product. I have a question:

My question: Distributing $N$ points on the sphere so that the sum of their mutual distances is maximized?

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    $\begingroup$ why not just put $N/2$ points on the North pole and $N/2$ points on the South pole? $\endgroup$
    – Carlo Beenakker
    Commented Jul 11, 2018 at 9:47
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    $\begingroup$ Nice question !!!! $\endgroup$
    – zeraoulia rafik
    Commented Jul 11, 2018 at 12:58
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    $\begingroup$ @Carlo Beenakker For n=4 the tetrahedron is better. $\endgroup$
    – LeechLattice
    Commented Jul 11, 2018 at 13:18

1 Answer 1

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This is actually an open problem, the only proven optimal configurations I have found are the universal configurations with N=1,2,3,4,6,12, and the dipyramid with N=5. Some work for N=7 is done here, which is compatible with this. Some bounds are given in the paper Sums of distances between points of a sphere.

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