On the monotonicity of perimeter of convex bodies

Let $n\ge 2$ and let $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to [0,\mathrm{\infty })$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in ${\mathbb{R}}^n$, the monotonicity of anisotropic $\mathrm{\Phi }$-perimeters holds, i.e. $P_{\mathrm{\Phi }}(A)\le P_{\mathrm{\Phi }}(B)$. In this note, we prove a quantitative lower bound on the difference of the $\mathrm{\Phi }$-perimeters of $A$ and $B$ in terms of their Hausdorff distance.