Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets.

Let $P_S$ be the multivariate Normal conditioned on a ball of radius $d$ ?

Is Covariance$(P_S) \preceq I$ for all $d$?

Is the above true for any convex set S?

Is there a simple proof or counterexample?


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