Let $v$ be a given vector with $|v|_{Sigma^{-1}} leq 1$, where $Sigma$ is a positive semi-definite matrix and $|v|_{Sigma^{-1}} = sqrt{v^topSigma v}$. Meanwhile, let $u$ be a random vector drawn from $N(0,Sigma^{-1})$. We know that for any given vector $phi$, it holds that

$$

P(phi^top u > phi^top v) > c

$$

where $c$ is a given positive absolute constant.

Question: Does a similar anti-concentration property still hold when we additionally condition on the event

$

mathcal{E} = {u in mathcal{C}}

$, where $mathcal{C}$ is a given set such that $v in mathcal{C}$ and moreover $v$ is in the interior of $mathcal{C}$? In other words, does it hold that

$$

P(phi^top u > phi^top v ,|, mathcal{E})> c’,

$$

where $c’$ is a given positive absolute constant? One particular example of $mathcal{C}$ that I am interested in is

$$

mathcal{C} = {u: | u + w | leq 1 },

$$

where $w$ is a given vector such that $| v + w | leq 1-delta$ with $delta > 0$.