Calculus of Variations and Geometric Measure Theory

F. Giannetti - G. Stefani

On the monotonicity of non-local perimeter of convex bodies

created by stefani on 20 Sep 2023


Submitted Paper

Inserted: 20 sep 2023
Last Updated: 20 sep 2023

Year: 2023


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.

Keywords: Hausdorff distance, monotonicity, convex body, non-local perimeter, Schwartz symmetrization