  Under mild assumptions on the kernel $K\ge0$, 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.
