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Suppose $u_{10}, u_{20}$ are two fixed orthonormal vectors in $\mathbb R^D$. $u_1, u_2$ are also two orthonormal vectors in $\mathbb R^D$ and they are individually uniformly distributed on the unit sphere. Consider the following set $$V = \{(u_1, u_2):a <(u_1'u_{10})^2 + (u_2'u_{20})^2 + (u_2'u_{10})^2 + (u_1'u_{20})^2< b\}.$$ We want to bound $Vol(V)$ where the upper bound $f(a, b)$ and lower bound $g(a, b)$ should involve same power of $a, b$.

Note that if I had only one variable $u$ and correspondingly instead of $u_{10}$ and $u_{20}$ I had only $u_0$, I could've obtained bounds using cylinders: the surface area can be easily bounded by the cylinders with the bases $u'u_0 = b$ and $u'u_0 = a$. Any suggestion would be great!

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  • $\begingroup$ I don't know what you mean by $u_1'$. $\endgroup$ Feb 12 at 8:25
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    $\begingroup$ Oh it’s the transpose of $u_1$. $\endgroup$ Feb 12 at 9:06

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