function $\theta \mapsto
M_f(\theta, t)$ is $C$-Lipshitz on $S^{n-1}$. Let $E \in G_{n,
\ell}$ be a random subspace, uniformly distributed in $G_{n,
\ell}$. We would like to apply Proposition \[dvoretzky\] with $k
= \ell, L = C$ and $\eps = \frac{1}{2} n^{-\delta/2}$. Note that for this choice of parameters, $$k = \ell \leq c \delta A^{-1} \log n \leq \hat{c} \eps^2 (\log
1/\eps)^2 n \ \ \ \text{and} \ \ \ 2 \eps L \leq e^{-2 c
\delta \log n} \leq e^{-2 A \ell},$$ provided that $c$ is a sufficiently small, positive universal constant, and that $n$ is greater than some universal constant. Hence the appeal to Proposition \[dvoretzky\] is legitimate. From the conclusion of that proposition, with probability larger than $1 - e^{-c^{\prime} n^{1-\delta}}$ of selecting $E$, $$\label{eq_434} \sup_{\theta \in S^{n-1} \cap E} M_f(\theta,t) \leq
 e^{-2 A \ell}  + \inf_{\theta \in S^{n-1} \cap E} M_f(\theta, t).$$ For any fixed $t \in \RR$, the estimate (\[eq\_434\]) holds with probability greater than $1 - e^{-c^{\prime} n^{1-\delta}}$. Denote $I = \{ i \cdot e^{-2 A \ell} \, ; \, i=-\lceil e^{30 A
\ell} \rceil,...,\lceil e^{30 A \ell} \rceil \}$. Then, with probability greater than $1 - e^{ -\bar{c} n^{1-\delta}}$, we obtain $$\forall 