Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. 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 positive absolute constant that depends on $v$.

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 positive absolute constant that depends on $v$ and $\mathcal{C}$? 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 $\delta > 0$.

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. 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 positive absolute constant that depends on $v$.

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 positive absolute constant that depends on $v$ and $\mathcal{C}$? 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 $\delta > 0$ (e.g., $\delta = 0.01$).

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. 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 that depends on $v$.

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 that depends on $v$ and $\mathcal{C}$? 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 $\delta > 0$.

Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. 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 positive absolute constant that depends on $v$.

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 positive absolute constant that depends on $v$ and $\mathcal{C}$? 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 $\delta > 0$.