On the Monotonicity of Perimeter of Convex Bodies

Let

and let

\Phi\colon{\mathbb R}^n\to[0,\infty) Φ  ⁣ : R n → [ 0 , ∞ )

be a positively

1 1

-homogeneous and convex function. Given two convex bodies

A\subset B A ⊂ B

in

{\mathbb R}^n R n

, the monotonicity of anisotropic

\Phi Φ

-perimeters holds, i.e.

P_\Phi(A)\le P_\Phi(B) P Φ ( A ) ≤ P Φ ( B )

. In this note, we prove a quantitative lower bound on the difference of the

\Phi Φ

-perimeters of

A A

and

B B

in terms of their Hausdorff distance.