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Let $B$ be the unit ball in $\mathbb{R}^n$, and let $c\in(0,1)$ be a constant. I'm trying to find the asymptotics for the volume of the intersection $[\frac{c}{\sqrt{n}},1]^n\cap B$ as $n\rightarrow\infty$. Is there nice asymptotic formula for this volume?

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    $\begingroup$ Depends on the precision you want. For the crude asymptotic just think Gaussian. $\endgroup$
    – fedja
    Commented Jul 26, 2023 at 9:38
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    $\begingroup$ In other words, it is $(2\pi)^{n/2}n^{-(n+1)/2}Q(c)^n$ where $Q(c)=\min_{t>0}[\Phi(ct)\frac 1t e^{t^2/2}]$ and $\Phi(z)=P(\xi>z)$ ($\xi$ is the standard 1D normal r.v.) is the Gaussian tail function. That is correct up to a constant factor depending on $c\in(0,1)$ but not on $n$. $\endgroup$
    – fedja
    Commented Jul 26, 2023 at 11:49

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