  Let $n\ge2$ and let $\Phi\colon\mathbb{R}^n\to[0,\infty)$ be a positively
$1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in
$\mathbb{R}^n$, the monotonicity of anisotropic $\Phi$-perimeters holds, i.e.
$P_\Phi(A)\le P_\Phi(B)$. In this note, we prove a quantitative lower bound on
the difference of the $\Phi$-perimeters of $A$ and $B$ in terms of their
Hausdorff distance.
