on $A$ and $U$, and since $U \leq A$, we can reduce the dependence of such constant on $A$ only.

$\Box$

Appendix 6: Anti-concentration and comparison bounds for maxima of Gaussian random vectors and Berry-Esseen bounds for polyhedral sets {#app:high.dim.clt}
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Now we collect some results that can be are derived from [@chernozhukov2015comparison], [@cherno2] and [@nazarov1807maximal]. However, our statement of the results is slightly different than in the original papers. The reason for this is that we need to keep track of some constants in the proofs that affect our rates.

The following anti-concentration result for the maxima of Gaussian vectors follows from Lemma A.1 in [@cherno2] and relies on a deep result in [@nazarov1807maximal].

\[thm:anti.concentration\] Let $(X_1\ldots,X_p)$ be a centered Gaussian vector in $\mathbb{R}^p$ with $\sigma_j^2 = \mathbb{E}[X_j^2] > 0$ for all $j=1,\ldots,p$. Moreover, let $\underline{\sigma} = \min_{1 \leq j \leq p} \sigma_j$. Then, for any $y = (y_1,\ldots,y_p) \in \mathbb{R}^p$ and $a > 0$ $$\mathbb{P}(  X_j  \leq y_j + a, \forall j) - \mathbb{P}( X_j
        \leq y_j, \forall j) \leq \frac{a}{\underline{\sigma}} \left( \sqrt{2 \log p} + 2 \right).$$

The previous result implies that, for any $a > 0$ and $y = (y_1,\ldots,y_p) \in
\mathbb{R}^p_+$, $$\mathbb{P}(  |X_j|  \leq y_j + a, \forall j) - \mathbb{P}( |X_j|
        \leq y_j, \forall j) \leq \frac{a}{\underline{\sigma}} \left(
        \sqrt{2 \log 2p} + 2 \right)$$ and that, for any $y > 0$, $$\mathbb{P}( \max_j |X_j | \leq y + a) - \mathbb{P}(\max_j |X_j|
        \leq y) \leq \frac{a}{\underline{\sigma}} \left( \sqrt{2 \log 2p} + 2 \right).$$

The following high-dimensional central limit theorem follows from Proposition 2.1 in [@cherno2] and . Notice that we have kept the dependence on the minimal variance explicit.

\[thm:high.dim.clt\] Let $X_1,\ldots,X_n$ be independent centered 