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Let $n$ be a positive integer, $\sigma > 0$, and $h > 0$. Consider a height-$h$ piece of the standard $n$-dimensional sphere, defined by $$ S_n(h) := \{x \in \mathbb R^n \mid \|x\|_2 \le 1,\;x_1 \ge h\}. $$

Question. What is the value of the integral $$I_{n,\sigma}(h):=\mathbb P_{x \sim \mathcal N(0,\sigma^2 I)}(x \in S_n(h)) = (2\pi\sigma^2)^{-n/2}\int_{S_n(h)}e^{-\frac{1}{2\sigma^2}\|x\|^2}dx $$

Observations

  • The case, $h=0$ which corresponds to the Gaussian measure of a hemi-sphere, is well-known (see here, for example) to equal $$ I_{n,\sigma}(0) = \frac{1}{2}\frac{\gamma(n/2,1/(2\sigma^2))}{\Gamma(n/2)} \sim \frac{1}{2}\sqrt{\sigma n}, $$ where $\gamma(s, x) := \int_{0}^x e^{-t}t^{s-1}dt$ is the lower incomplete gamma function and $\Gamma(s):=\gamma(s,\infty)$ is the usual gamma function.
dohmatob
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    The segment your are considering is a spherical cap of an n-dimensional hypersphere. See my answer in this post for calculating the surface area of this spherical cap, with a reference therein, which will help you with your problem: https://math.stackexchange.com/questions/2238156/ – Andreas Jan 09 '20 at 17:21
  • Thanks for the ref. But in my case this would lead to complicated integrals (complicated in the sense that I'm not quite sure how simplify them using any special functions with know properties) of the kind here https://math.stackexchange.com/q/3505099/168758 – dohmatob Jan 11 '20 at 19:17

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