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What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?

The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points. All the points are on a single sphere of radius $R$.

Here's a visualization: enter image description here

Acknowledgements: Based on this Physics S.E. question. Image from StackOverflow.

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raindrop
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  • Mean distance between 2 points within a sphere http://math.stackexchange.com/questions/167932/mean-distance-between-2-points-within-a-sphere might help – raindrop Sep 05 '12 at 04:17
  • I seem to recall hearing that the problem of maximizing the minimum is unsolved. But maximizing the sum is another matter. – Michael Hardy Sep 05 '12 at 04:24
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    A bunch of previous questions are closely related. None is an exact duplicate of this one, but you may find the answers and references there of interest. –  Sep 05 '12 at 04:56
  • "Straight line"? Through the sphere or on the surface? – Henry Sep 05 '12 at 06:37
  • Straight line through the sphere. – raindrop Sep 06 '12 at 16:43
  • Are you looking for an analytical expression of the configuration or a numerical solution? This is a nonconvex optimization problem and I suspect it has several local solutions that are not global maximizers. – Dominique Mar 06 '13 at 23:10
  • analytical, I just want to understand the steps involved in solving the problem – raindrop Mar 07 '13 at 22:20
  • You may be able to find an analytical expression if you measure geodesic distance (on the surface of the sphere) and use spherical coordinates. This is just a suggestion that could put you on track for the harder question you ask. – Dominique Aug 18 '13 at 17:17

1 Answers1

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As far as I know the answer to the general question is unknown. For the computer approach you can look at this article by Berman and Hanes. Here it is shown that the result for 5 points on the sphere can be found in finite time by computer. Also you can find some interesting references in the introduction part.

Hope, this will help

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