# pr.probability – Anti-concentration of Gaussian When Conditioning on Event

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 $$delta > 0$$.