A bound for the perimeter of inner parallel bodies

TL;DR

This paper establishes a sharp, regularity-independent lower bound for the perimeter of inner parallel bodies of convex sets, based solely on the original perimeter and inradius, with conditions for equality.

Contribution

It provides a new, precise lower bound for the perimeter of inner parallel bodies of convex sets, correcting previous errors and clarifying equality conditions.

Findings

The bound depends only on perimeter and inradius.

The bound is sharp and independent of boundary regularity.

Conditions for equality are explicitly characterized.

Abstract

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body $\Omega$. The bound depends only on the perimeter and inradius $r$ of the original body and states that \[|\partial\Omega_t| \geq \Bigl(1-\frac{t}{r}\Bigr)^{n-1}_+ |\partial \Omega|.\] In particular the bound is independent of any regularity properties of $\partial\Omega$. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes. This is a revised version of the paper published in J. Funct. Anal. (2016) where an error is addressed in accordance with a corrigendum to appear in J. Funct. Anal. The main result of the paper remains the same but an error in Lemma 2.1 has been corrected and the subsequent proofs have been adapted accordingly.