  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.
