function. Fix $\theta \in S^{n-1}$. Then,

1.  For $t \geq 0$ we have $
     1 - 2 e^{-|t|/10} \leq M_f(\theta, t) \leq 1$.

2.  For $t \leq 0$ we have $
    0 \leq M_{f} \left( \theta, t \right) \leq 2 e^{-|t|/10}.$

\[lem\_1028\]

*Proof:* Let $X$ be a random vector with density $f$. Then $\EE |\langle X, \theta \rangle|^2 = 1$. We use Lemma \[lem\_1010\](i), with the function $F(x) = |\langle x,
\theta \rangle|$, to deduce the desired inequalities. $\square$

The space of all isotropic, log-concave functions in a fixed dimension is a compact space, with respect to, e.g., the $L^1$-metric. In particular, one-dimensional log-concave functions are quite rigid. For instance, suppose that $g: \RR \rightarrow
[0, \infty)$ is an isotropic, log-concave function. Then (see Hensley [@hensley] and also, e.g., [@lovasz Lemma 5.5] or [@fradelizi]), $$\frac{1}{10} \leq g(0) \leq \sup_{x \in \RR} g(x) \leq 1.
\label{eq_547}$$ We conclude that for any log-concave, isotropic function $f: \RR^n
\rightarrow [0, \infty)$, $$\left|  M_f(\theta, t)  - M_f(\theta, s)  \right| \leq |t - s| \ \
\ \text{for all} \ s, t \in \RR, \ \theta \in S^{n-1}.
\label{eq_547_}$$ To prove (\[eq\_547\_\]), we set $E = \RR \theta$ and $g =
\pi_E(f)$. Then $g$ is 