On the monotonicity of non-local perimeter of convex bodies

Under mild assumptions on the kernel $K\ge 0$, the non-local $K$-perimeter $P_K$ satisfies the monotonicity property on nested convex bodies, i.e., if $A\subset B\subset {\mathbb{R}}^n$ are two convex bodies, then $P_K(A)\le P_K(B)$. In this note, we prove quantitative lower bounds on the difference of the $K$-perimeters of $A$ and $B$ in terms of their Hausdorff distance, provided that $K$ satisfies suitable symmetry properties.