A bound for the perimeter of inner parallel bodies

Simon Larson

Published 2015-08-26Version 1

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 of the original body and states that \[ |\partial\Omega_t| \geq \Bigl(1-\frac{t}{R_{in}}\Bigr)^{n-1}_+ |\partial \Omega|. \] 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 rather straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes.