Let $S^{d-1}$ be the unit sphere (centered at $O$) in $d$ dimensions. One can show that when $d=3$, for fixed $x\in S^{2}$ the area of $P(x) = \{y\in S^{2}, \angle xOy < \alpha \}$ is $2\pi(1-\cos\alpha)$. What is the analogous result in $d$ dimensions? Any help appreciated.
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1Here's an approach using the incomplete beta function: https://scialert.net/fulltext/?doi=ajms.2011.66.70 – David K Jan 02 '19 at 18:31
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It looks like this was answered before (with the incomplete beta function) in an answer to https://math.stackexchange.com/questions/2238156/what-is-the-surface-area-of-a-cap-on-a-hypersphere, which (except for minor differences in notation) seems completely equivalent to this question. – David K Jan 03 '19 at 02:45