  We extend the isotonic analysis for Wicksell's problem to estimate a
regression function, which is motivated by the problem of estimating dark
matter distribution in astronomy. The main result is a version of the
Kiefer--Wolfowitz theorem comparing the empirical distribution to its least
concave majorant, but with a convergence rate $n^{-1}\log n$ faster than
$n^{-2/3}\log n$. The main result is useful in obtaining asymptotic
distributions for estimators, such as isotonic and smooth estimators.
