through $x$. The Prékopa-Leindler inequality (see [@Pr1], [@Le], [@Pr2] or the first pages of [@pisier_book]), which is a functional version of Brunn-Minkowski, implies that $\pi_E(f)$ is log-concave whenever $f$ is log-concave and integrable. Therefore, when $f$ is isotropic and log-concave, $\pi_E(f)$ is also isotropic and log-concave. A further consequence of the Prékopa-Leindler inequality, is that when $f$ and $g$ are integrable log-concave functions on $\RR^n$, so is their convolution $f * g$. (The latter result actually goes back to [@davidovic], [@leker] and [@schoenberg].)

Let $n \geq 1$ be an integer, and let $X$ be a random vector in $\RR^n$ with a log-concave density. Assume that $F: \RR^n \rightarrow [0, \infty)$ is an even, convex function, such that $F(t x) = t F(x)$ for all $t > 0,
x \in \RR^n$. Denote $E = \sqrt{\EE |F(X)|^2}$. Then,

1.  $\displaystyle
    Prob \left \{ F(X) \geq t E  \right
    \} \leq 2 e^{-t/10}$ for all $t \geq 0$.

Additionally, let $0 < \eps \leq \frac{1}{2}$, and let $M > 0$ satisfy $ Prob \{ F(X) \geq M \} \leq \eps. $ Then,

1.  $\displaystyle
    Prob \left \{ F(X) \geq t M \right \} \leq (1 - \eps)
    \left(\frac{\eps}{1-\eps} \right)^{(t + 1)/2}$ for all 