
Journal of Convex Analysis 25 (2018), No. 1, [final page numbers not yet available] 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 (396 KB)] for subscribers only. 