On the monotonicity of non-local perimeter of convex bodies

Under mild assumptions on the kernel K≥0, the non-local K-perimeter PK satisfies the monotonicity property on nested convex bodies; i.e.\ if A⊂ Bⁿ are two convex bodies, then PK(A)≤ PK(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.