  Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard
regularity conditions, we derive the asymptotic distribution of the supremum
norm of the difference between $f$ and its Grenander-type estimator on
sub-intervals of $[0,1]$. The rate of convergence is found to be of order
$(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.
