Order statistics of vectors with dependent coordinates

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb R^n$. We show that the random vector $Y=T(X)$ satisfies
$$ \mathbb{E} \sum_{j=1}^k j\min_{i\leq n}{X_{i}}^2 \leq C \mathbb{E} \sum_{j=1}^k j\min_{i\leq n}{Y_{i}}^2 $$

for all $k\leq n$, where "$j\min$" denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen–Loève basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.