On the monotonicity of non-local perimeter of convex bodies

TL;DR

This paper proves that the non-local $K$-perimeter of convex bodies is monotonic with respect to set inclusion and establishes quantitative bounds on perimeter differences based on Hausdorff distance, under symmetry assumptions on the kernel.

Contribution

It provides the first quantitative bounds on the perimeter difference for convex bodies under non-local perimeter measures with symmetry conditions.

Findings

Monotonicity of non-local $K$-perimeter for convex bodies.

Quantitative lower bounds on perimeter differences.

Results depend on symmetry properties of the kernel.

Abstract

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.