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This is a companion contrast to the earlier analogous question for unit $n$-cubes, where the answer (provided by several respondents) is $\infty$ .

What is the limit, as $n \to \infty$, of the expected distance between two points chosen uniformly at random within a unit-radius hypersphere in $\mathbb{R}^n$, i.e., in the unit-radius $n$-ball?

Dividing OEIS A093530 by A093531 I see that it appears to be approaching approximately $1.37$ for odd $n$, but I wonder if the limit is actually known, either exactly or to significant precision? I cannot quite extract an answer from the MathWorld article...

This certainly provides a dramatic contrast between the $n$-cube and the $n$-sphere!

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    $\begingroup$ I'm not sure there's such a contrast between cubes and spheres. In high dimensions a unit cube is vastly larger than a unit sphere, as measured by diameter or volume, so it's not as fair a comparison as the word "unit" suggests. $\endgroup$
    – Henry Cohn
    Commented Apr 15, 2014 at 0:44
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    $\begingroup$ If instead we look at the ratio of expected distance to diameter, then for spheres the limit is asymptotically $\frac{1}{\sqrt{2}}$, while for cubes it is (from Nate Eldredge's answer to the linked question) $\frac{1}{\sqrt{6}}$. Is it known whether Spheres maximize this ratio, whether in finite dimensions or asymptotically? $\endgroup$
    – Kevin P. Costello
    Commented Apr 15, 2014 at 21:15

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Both points will be very close to (let's pretend: on) the surface with prob almost 1. Call the first point the north pole. By concentration of measure for the sphere, a randomly chosen second point is almost guaranteed to be almost on the equator, so the limit should be $\sqrt{2}$.

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  • $\begingroup$ May I ask: Why can you assume that the first point is the north pole? Concentratoin of measure again? $\endgroup$
    – Joseph O'Rourke
    Commented Apr 15, 2014 at 0:17
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    $\begingroup$ I condition on the first point, and the sphere looks the same from any point. $\endgroup$
    – Christian Remling
    Commented Apr 15, 2014 at 0:18
  • $\begingroup$ Are you reading the question as "two points on the surface of the unit sphere", or ought this be overwhelmingly likely? $\endgroup$
    – usul
    Commented Apr 15, 2014 at 0:57
  • $\begingroup$ Yes, "sphere" (as opposed to "ball"), but even if it's the ball, almost all the volume is near the surface, so it won't matter. $\endgroup$
    – Christian Remling
    Commented Apr 15, 2014 at 1:02
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    $\begingroup$ I should have known, after your first comment (sorry for the fake discussion)... Solution stays the same, though, by also applying the trivial concentration of measure near the surface first. $\endgroup$
    – Christian Remling
    Commented Apr 15, 2014 at 1:24

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