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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 aprobability 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, inparticular, that for any isotropic log-concave measure $\mu$ one may find aconvex 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 withrespect to any log-concave measure with density $f$ in an appropriate position.  Our lower bound is attained for a class of distributions including thestandard normal distribution. Our upper bound is attained, say, for a uniformmeasure on the cube.  In addition, for isotropic log-concave measures we prove an upper bound oforder $n^2$ for the maximal $\mu$-perimeter of a convex set.

Some remarks about the maximal perimeter of convex sets with respect to probability measures