Imagine you have x blue points and x red points in two or more dimensions. Is it true that the sum of all the distances between points of the same color is never more than the sum of all the distances between points of different colors?
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The answer after the edit:
Your question is identical to this one in math.stackexchange, which contains a proof the answer is yes.
The answer to the original question before the edit (where the number of red and blue points can differ) is
No
Because
Imagine two blue points and zero red points.
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$\begingroup$ Argh.. sorry the question had a typo! $\endgroup$– SimdCommented Mar 22, 2018 at 16:19