\leq e^{-C_0 n} +
\inf_{\theta \in S^{n-1}} M_{f}(\theta, t) \ \ \ \text{for all} \
\ t \in \RR. \label{eq_848_}$$ Suppose that $Y$ is a random vector in $\RR^n$ with density $f$. Then for all $0 < \eps < 1$, $$Prob \left \{ \left| \, \frac{|Y|}{\sqrt{n}} -1 \, \right| \geq \eps
\right \} \leq C e^{-c \eps^2 n }. \label{eq_309}$$ \[cor\_735\]

For $n \geq 1$ and $v > 0$ we define $\gamma_{n,v}: \RR^n
\rightarrow [0, \infty)$ to be the function $$\gamma_{n,v}(x) = \frac{1}{(2 \pi v)^{n/2}} \exp \left(
-\frac{|x|^2}{2v} \right). \label{gamma_v}$$ Then $\gamma_{n,v}$ is the density of a gaussian random vector in $\RR^n$ with expectation zero and covariance matrix that equals $v Id$, where $Id$ is the identity matrix. We write $O(n)$ for the group of orthogonal transformations of $\RR^n$.

Let $n \geq 1$ be an integer, let $\alpha \geq 5$, and let $f: \RR^n \rightarrow [0, \infty)$ be an isotropic, log-concave function. Assume that $$\sup_{\theta \in S^{n-1}} M_{f}(\theta, t) \leq e^{-5 \alpha n} +
\inf_{\theta \in S^{n-1}} M_{f}(\theta, t) \ \ \ \text{for all} \
\ t \in \RR. \label{eq_848}$$ Denote $g = f * \gamma_{n, 1}$, where $*$ stands for convolution. Then, $$\sup_{\theta \in S^{n-1}} g(t \theta) \leq e^{-\alpha 