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Consider $n$ points chosen uniformly and independently on the surface of a sphere with circumference $1$ at the equator. What is the expected minimum (great-circle) distance between any pair of points?

If we considered a circle instead the answer would be $1/n^2$.

  • well there are $\lceil{n-2\over 2}\rceil+2 $ points in a hemisphere.... –  Jan 25 '20 at 13:38
  • A sphere doesn't have a circumference. – joriki Jan 25 '20 at 14:10
  • @joriki https://www.calculatorsoup.com/calculators/geometry-solids/sphere.php. What is the correct term? –  Jan 25 '20 at 14:13
  • OK, I reopened due to the difference. However, note that the asymptotics will be the same and a closed-form solution is similarly unlikely, so the answers will probably not differ. – joriki Jan 25 '20 at 14:17
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    I'd call it the circumference of (or perhaps at) the equator, or generally the circumference of the great circles of the sphere. – joriki Jan 25 '20 at 14:18

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