Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},p_Y)$. Define distance measure $d: \mathbb{R}^d\times \mathbb{R}^d \mapsto \mathbb{R} $ as $d(x,y) = (x-y)^T(x-y)$, where $T$ is the transpose. Consider a linear projection $Q: \mathbb{R}^d \mapsto \mathbb{R}^k$ for $k \gg d$.
Under which condition on $p_X$, $p_Y$, and $Q$, we can find a lower bound on the probability of the following event: $$\dfrac{d(Y,E[X])}{d(X,E[X])} \leq \dfrac{d(QY,QE[X])}{d(QX,QE[X])}$$ where $E[X]$ is the expected value of $X$.
