$|t| < 20 A \ell$. Thus (\[eq\_618\]) holds for all $t \in \RR$, under the assumption that $E$ satisfies (\[eq\_443\]).

Recall that $\theta_1, \theta_2 \in S^{n-1} \cap E$ are arbitrary, hence we may take the supremum over $\theta_1$ and the infimum over $\theta_2$ in (\[eq\_618\]). We discover that whenever the subspace $E$ satisfies (\[eq\_443\]), it necessarily also satisfies (\[eq\_616\]). The probability for a random $\ell$-dimensional subspace $E \subset \RR^n$ to satisfy (\[eq\_443\]) was shown to be greater than $1 - e^{-\bar{c} n^{1
-\delta}}$. The lemma thus follows. $\square$

*Remark.* For the case where $f$ is even, Lemma \[lem\_dvo\] follows from a direct application of Dvoretzky’s theorem in Milman’s form. Indeed, in this case, $\theta \mapsto |\theta| M_f(\theta,
t)^{-1}$ is a norm, and Lemma \[lem\_dvo\] asserts that this norm is almost Hilbertian when restricted to certain random subspaces.

Almost spherical log-concave functions {#section4}
======================================

A large portion of this section is devoted to proving the following proposition.

There exist universal constants $C_0,C > 1$ and $0 < c < 1$ for which the following holds: Let $n \geq 1$ be an integer and let $f: \RR^n \rightarrow [0, \infty)$ be an isotropic, log-concave function. Assume that $$\sup_{\theta \in S^{n-1}} M_{f}(\theta, t) 