t \in I, \ \ \ \sup_{\theta \in S^{n-1} \cap E} M_{f}
(\theta, t) \leq e^{-2 A \ell} +  \inf_{\theta \in S^{n-1} \cap E}
M_{f} (\theta, t). \label{eq_443}$$ Indeed, the estimate for the probability follows from the inequality $(2 e^{30 A \ell} + 3) e^{ - c^{\prime} n^{1-\delta}}
\leq e^{-\bar{c} n^{1-\delta}}$.

Fix an $\ell$-dimensional subspace $E
\subset \RR^n$ that satisfies (\[eq\_443\]). Select $\theta_1, \theta_2 \in S^{n-1} \cap E$. We will demonstrate that for any $t \in \RR$, $$M_{f} (\theta_1, t) \leq e^{-A \ell} +
M_{f} (\theta_2, t).
\label{eq_618}$$ To that end, note that when $|t| \geq 20 A \ell$, by Lemma \[lem\_1028\], $$\left| M_f(\theta_1, t) - M_f (\theta_2, t) \right| \leq 2 e^{-|t| / 10}
\leq 2 e^{-2 A \ell} \leq e^{-A \ell}.
\label{eq_626}$$ Hence (\[eq\_618\]) holds for $|t| \geq 20 A \ell$. We still need to consider the case where $|t| < 20 A \ell$. In this case, $|t|
\leq e^{20 A \ell}$ and hence there exists $t_0 \in I$ with $|t -
t_0| \leq \frac{1}{2} \cdot e^{-2 A \ell}$. According to (\[eq\_547\_\]) from Section \[section2\], the function $t
\mapsto M_{f} (\theta_i, t)$ is $1$-Lipshitz for $i=1,2$. Therefore, by using (\[eq\_443\]), we conclude (\[eq\_618\]) also for the case where 