Calculus of Variations and Geometric Measure Theory
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G. Stefani

On the monotonicity of perimeter of convex bodies

created by stefani on 01 Dec 2016
modified on 04 Jan 2018


Accepted Paper

Inserted: 1 dec 2016
Last Updated: 4 jan 2018

Journal: Journal of Convex Analysis
Volume: 25
Number: 1
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: Convex body, anisotropic perimeter, Hausdorff distance, Wulff inequality


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