Let $S^{d-1}=\{(x_1,\cdots,x_{d})\in {\mathbb R}^{d}|{x_1}^2+\cdots+{x_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ 'a great circle'. Also, let us call one of two components gotten by cutting $S^{d-1}$ by a great circle '$d-1$ dimensional semi-spherical surface'. Also, suppose that a point set on $S^{d-1}$ is semi-spherical when it is included in a $d-1$ dimensional semi-spherical surface.

Then, here is my question.

Question : Letting $d\ge 2$, find the probability $f(d,n)$ that $n$ points which are randomly selected from $S^{d-1}$ are semi-spherical.

Remark : Suppose that $n$ points are selected randomly with equable probability.

Example : We can easily get $f(2,n)=n\cdot 2^{1-n}.$

Motivation : I've known the $d=2$ case, but I'm facing difficulty for $d$ in general. Can anyone help?


2 Answers 2


This is answered in painstaking detail here.

  • $\begingroup$ Wow! Thank you very much for great info. $\endgroup$
    – mathlove
    Oct 23, 2013 at 17:09

Is this not the probability that if you toss a coin n times then you get either n heads or n tails? Paul

  • 1
    $\begingroup$ Nope, it is not. $\endgroup$
    – Igor Rivin
    Oct 23, 2013 at 16:59
  • 2
    $\begingroup$ I think the question is looking for the probability that the points are in some hemisphere (possibly at a funny angle), not necessarily one of two fixed hemispheres corresponding to a single great circle. $\endgroup$
    – Henry Cohn
    Oct 23, 2013 at 17:03

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