On the monotonicity of perimeter of convex bodies

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.