$$|g(\theta) - M| \leq \eps L \ \ \ \ \text{for all} \ \theta \in
S^{n-1} \cap E. \label{eq_700}$$ Here, $0 < c,\hat{c} < 1$ are universal constants. \[dvoretzky\]

Our use of “Dvoretzky’s theorem type” arguments in the next lemma is inspired by the powerful methods of Paouris in [@Pa1], [@Pa2], [@Pa3].

Let $n \geq 1$ be an integer, let $A \geq 1, 0 < \delta \leq \frac{1}{2}$ and let $f: \RR^n \rightarrow [0, \infty)$ be an isotropic, log-concave function. Assume that $1 \leq \ell \leq c \delta
A^{-1} \log n$ is an integer, and let $E$ be a random $\ell$-dimensional subspace in $\RR^n$. Then with probability greater than $1 - e^{-c n^{1-\delta}}$, $$\sup_{\theta \in S^{n-1} \cap E} M_{f}(\theta, t) \leq e^{-A \ell}
+ \inf_{\theta \in S^{n-1} \cap E} M_{f}(\theta, t) \ \ \
\text{for all} \ \ t \in \RR. \label{eq_616}$$ Here, $0 < c < 1$ is a universal constant. \[lem\_dvo\]

*Proof:* We may assume that $n$ exceeds a given universal constant, since otherwise, for a suitable choice of a small universal constant $c$, there is no $\ell$ with $1 \leq
 \ell
\leq c \delta A^{-1} \log n$. Fix a real number $t$. According to Proposition \[lem\_bobkov\], the 