  In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$
with respect to various classes of measures. Firstly, we show that for a
probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions,
there exists a convex set of $\mu$-perimeter at least
$C\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}.$ This implies, in
particular, that for any isotropic log-concave measure $\mu$ one may find a
convex set of $\mu$- perimeter of order $n^{\frac{1}{8}}$.
  Secondly, we derive a general upper bound of $Cn||
f||^{\frac{1}{n}}_{\infty}$ on the maximal perimeter of a convex set with
respect to any log-concave measure with density $f$ in an appropriate position.
  Our lower bound is attained for a class of distributions including the
standard normal distribution. Our upper bound is attained, say, for a uniform
measure on the cube.
  In addition, for isotropic log-concave measures we prove an upper bound of
order $n^2$ for the maximal $\mu$-perimeter of a convex set.
