  Let $I$ be a homogeneous ideal in a polynomial ring $S$. In this paper, we
extend the study of the asymptotic behavior of the minimum distance function
$\delta_I$ of $I$ and give bounds for its stabilization point, $r_I$, when $I$
is an $F$-pure or a square-free monomial ideal. These bounds are related with
the dimension and the Castelnuovo--Mumford regularity of $I$.
