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Suppose I have two hyperspheres in $n$ dimensions with radius $r$ each, whose centers are $d$ distance apart. Assuming $d\leq 2r$, the two hyperspheres intersect and the region consists of two spherical caps. I am interested in how the volume of the spherical cap behaves for constant $r$ and $d$ but as $n\rightarrow \infty$.

The wikipedia article (https://en.wikipedia.org/wiki/Spherical_cap) mentions an asymptotic result that for $n \rightarrow \infty$ but $\frac{\sqrt{n}d}{r}=$constant, $Vol(cap) \approx 2Vol(sphere)\left(1-F\left(\frac{\sqrt{n}d}{r}\right)\right)$. The citation references a paper which is in Russian, which I cannot read.

I am wondering if $Vol(cap) \sim 2Vol(sphere)\left(1-F\left(\frac{\sqrt{n}d}{r}\right)\right)$ for constant $r$ and $d$ but for large enough $n$. A proof sketch of why it is true (if it indeed is) or if it is not true, would be very appreciated. Thanks!

Another Grad student
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  • you mean $d\le 2r$? – leonbloy Aug 20 '20 at 15:30
  • have you seen this ? https://math.stackexchange.com/questions/162250/how-to-compute-the-volume-of-intersection-between-two-hyperspheres – leonbloy Aug 20 '20 at 15:31
  • Yes, $d\leq 2r$ thanks! Yes I saw this, the answer cites the same wikipedia article but perhaps forgets the detail (in the wiki) that $\frac{\sqrt{n}d}{r}=$constant. So I am wondering what is stated in the asnwer is indeed true or not. – Another Grad student Aug 20 '20 at 15:48
  • I would guess that, in your case, that implies that the ratio $Vol(cap)/Vol(Sphere) \to 0$, because $\frac{\sqrt{n}d}{r} \to + \infty$ – leonbloy Aug 21 '20 at 11:48

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