On the monotonicity of perimeter of convex bodies

Abstract

Let $n\ge 2$ and let $\Phi :{\mathbb{R}}^n\to [0,\infty )$ 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.

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