Calculus of Variations and Geometric Measure Theory

G. Stefani

On the monotonicity of perimeter of convex bodies

created by stefani on 01 Dec 2016
modified on 03 Oct 2021


Published Paper

Inserted: 1 dec 2016
Last Updated: 3 oct 2021

Journal: J. Convex Anal.
Volume: 25
Number: 1
Pages: 93--102
Year: 2018

ArXiv: 1612.00295 PDF


Let $n\ge2$ and let $\Phi\colon\mathbb{R}^n\to[0,\infty)$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in $\mathbb{R}^n$, the monotonicity of anisotropic $\Phi$-perimeters holds, i.e. $P_\Phi(A)\le P_\Phi(B)$. In this note, we prove a quantitative lower bound on the difference of the $\Phi$-perimeters of $A$ and $B$ in terms of their Hausdorff distance.

Keywords: anisotropic perimeter, Hausdorff distance, Wulff inequality, convex body