Geodesic
On the Monotonicity of Perimeter of Convex Bodies
Journal of convex analysis, Tome 25 (2018) no. 1, pp. 93-102
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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.
Classification : 52A20, 52A40
Mots-clés : Convex body, anisotropic perimeter, Hausdorff distance, Wulff inequality 
@article{JCA_2018_25_1_JCA_2018_25_1_a5,
     author = {G. Stefani},
     title = {On the {Monotonicity} of {Perimeter} of {Convex} {Bodies}},
     journal = {Journal of convex analysis},
     pages = {93--102},
     year = {2018},
     volume = {25},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2018_25_1_JCA_2018_25_1_a5/}
}
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G. Stefani. On the Monotonicity of Perimeter of Convex Bodies. Journal of convex analysis, Tome 25 (2018) no. 1, pp. 93-102. http://geodesic.mathdoc.fr/item/JCA_2018_25_1_JCA_2018_25_1_a5/