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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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Mean distance between 2 points within a sphere… might help – raindrop Sep 5 '12 at 4:17
I seem to recall hearing that the problem of maximizing the minimum is unsolved. But maximizing the sum is another matter. – Michael Hardy Sep 5 '12 at 4:24
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. – Rahul Sep 5 '12 at 4:56
"Straight line"? Through the sphere or on the surface? – Henry Sep 5 '12 at 6:37
Straight line through the sphere. – raindrop Sep 6 '12 at 16:43

2 Answers 2

up vote 1 down vote accepted

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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this is my anology. If something is something wrong please reply to me. I think that the lengths will add up to be maximum

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An easy counterexample is a $n=6$, where an octahedron has larger sum of distances than a hexagon. – Rahul May 30 '14 at 0:56

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