On the monotonicity of weighted perimeters of convex bodies

Abstract

We prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially-weighted anisotropic densities.

arXiv comment7 pages

References in corpus (9)

• Isoperimetry and stability properties of balls with respect to nonlocal energies
• On the isoperimetric problem with double density
• The $\epsilon -{\epsilon }^{\beta }$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets
• Weighted Cheeger sets are domains of isoperimetry
• The Cheeger problem in abstract measure spaces
• On the Steiner property for planar minimizing clusters. The anisotropic case
• On the Steiner property for planar minimizing clusters. The isotropic case
• Range convergence monotonicity for vector measures and range monotonicity of the mass
• On the convex components of a set in ${\mathbb{R}}^n$

Cited by in corpus (1)

• Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications