Anisotropic perimeter and isoperimetric quotient of inner parallel bodies

TL;DR

This paper provides a concise proof of existing results on the perimeter and isoperimetric quotient of inner parallel bodies and extends these findings to anisotropic cases, enhancing understanding of geometric properties under different metrics.

Contribution

It offers a simplified proof of known theorems and generalizes them to anisotropic settings, broadening their applicability in convex geometry.

Findings

Perimeter bounds for inner parallel bodies

Monotonicity of isoperimetric quotient under eikonal abrasion

Extension of results to anisotropic convex bodies

Abstract

The aim of this note is twofold: to give a short proof of the results in [S. Larson, A bound for the perimeter of inner parallel bodies, J. Funct. Anal. 271 (2016), 610-619] and [G. Domokos and Z. L\'angi, The isoperimetric quotient of a convex body decreases monotonically under the eikonal abrasion model, Mathematika 65 (2019), 119-129]; and to generalize them to the anisotropic case.