
Journal of Convex Analysis 25 (2018), No. 1, 093102 Copyright Heldermann Verlag 2018 On the Monotonicity of Perimeter of Convex Bodies Giorgio Stefani Scuola Normale Superiore, Piazza Cavalieri 7, 56126 Pisa, Italy giorgio.stefani@sns.it [Abstractpdf] 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 [ Fulltextpdf (107 KB)] for subscribers only. 