  We derive a lower bound on the differential entropy of a log-concave random
variable $X$ in terms of the $p$-th absolute moment of $X$. The new bound leads
to a reverse entropy power inequality with an explicit constant, and to new
bounds on the rate-distortion function and the channel capacity.
  Specifically, we study the rate-distortion function for log-concave sources
and distortion measure $| x - \hat x|^r$, and we establish that the difference
between the rate distortion function and the Shannon lower bound is at most
$\log(\sqrt{\pi e}) \approx 1.5$ bits, independently of $r$ and the target
distortion $d$. For mean-square error distortion, the difference is at most
$\log (\sqrt{\frac{\pi e}{2}}) \approx 1$ bits, regardless of $d$.
  We also provide bounds on the capacity of memoryless additive noise channels
when the noise is log-concave. We show that the difference between the capacity
of such channels and the capacity of the Gaussian channel with the same noise
power is at most $\log (\sqrt{\frac{\pi e}{2}}) \approx 1$ bits.
  Our results generalize to the case of vector $X$ with possibly dependent
coordinates, and to $\gamma$-concave random variables. Our proof technique
leverages tools from convex geometry.
