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Journal of Convex Analysis 25 (2018), No. 1, 093--102
Copyright Heldermann Verlag 2018

On the Monotonicity of Perimeter of Convex Bodies

Giorgio Stefani
Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy


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.

MSC: 52A20; 52A40

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