) \right\},$$ where, for $j=1,\ldots,s$, $t_{2j-1} = t_{2j} = \frac{t}{\|G_j\|}$.

Recalling that $\widehat{\nu} = G \widehat{\psi}$, we have that $$\left\|     \sqrt{n}(\hat{\nu} - \nu ) \right\|_\infty \leq t \quad
\text{if and only if } \quad 
\sqrt{n} (\hat{\psi} - \psi) \in P(G,t).$$ Similarly, if $\tilde{Z}_n \sim
N_b(0,V)$ and $Z_n  = G \tilde{Z}_n \sim N_s(0,\Gamma)$ $$\| Z_n \|_\infty \leq t\quad
\text{if and only if } \quad \tilde{Z}_n \in P(G,t).$$

Now consider the class $\mathcal{A}$ of all subsets of $\mathbb{R}^b$ of the form specified in , where $t$ ranges over the positive reals and $G$ ranges in $ \{  G(\psi(P)),  P \in \mathcal{P}\}$. Notice that this class is comprised of polytopes with at most $2s$ facets. Also, from the discussion above, $$\label{eq:simple.convex}
\sup_{ P \in \mathcal{P}_n} \sup_{t >0}  \left |  \mathbb{P}\left( \| \sqrt{n} (\hat{\nu} - \nu) \|_\infty
\leq t\right)  -\mathbb{P}\left( \| Z_n \|_\infty \leq t  \right) \right|  = 
\sup_{A \in \mathcal{A}} \left |  \mathbb{P}(\sqrt{n} (\hat{\psi} - \psi) \in A)- \mathbb{P}( \tilde{Z}_n \in A) \right|.$$

The claimed result follows from applying the Berry-Esseen bound for polyhedral classes, in the appendix, due to [@cherno2] to the term on the left hand side of . To that end, we need to ensure that conditions (M1’), (M2’) and (E1’) in that Theorem are satisfied.

For each $i=1,\ldots,n$, set $\tilde{W}_i=
(\tilde{W}_{i,1}, \ldots, \tilde{W}_{i,2s})= \left( (W_i - \psi)^\top v,v \in \mathcal{V}(G) \right)$. Condition (M1’) holds since, for each $l=1,\ldots,2s$, $$\mathbb{E}\left[ \tilde{W}^2_{i,l} \right] \geq \min_l v_l^\top V v_l \geq
    \lambda_{\min}(V),$$ where $V = \mathrm{Cov}[W]$. Turning to condition (M2’), we have that, for for each $l=1,\ldots,2s$ and $k=1,2$, $$\begin{aligned}
\mathbb{E}\left[ | \tilde{W}_{i,l}|^{2+k} \right]  & \leq \mathbb{E}\left[ |v_l^\top
(W_i - \psi)|^2  \|W_i - \psi\|^{k} \right]\\
& \leq   \mathbb{E}\left[ |v_l^\top (W_i - \psi)|^2 \right] \left( 2A \sqrt{b}
\right)^k\\
& \leq \ove