  Consider the problem of estimating the $\gamma$-level set
$G^*_{\gamma}=\{x:f(x)\geq\gamma\}$ of an unknown $d$-dimensional density
function $f$ based on $n$ independent observations $X_1,...,X_n$ from the
density. This problem has been addressed under global error criteria related to
the symmetric set difference. However, in certain applications a spatially
uniform mode of convergence is desirable to ensure that the estimated set is
close to the target set everywhere. The Hausdorff error criterion provides this
degree of uniformity and, hence, is more appropriate in such situations. It is
known that the minimax optimal rate of error convergence for the Hausdorff
metric is $(n/\log n)^{-1/(d+2\alpha)}$ for level sets with boundaries that
have a Lipschitz functional form, where the parameter $\alpha$ characterizes
the regularity of the density around the level of interest. However, the
estimators proposed in previous work are nonadaptive to the density regularity
and require knowledge of the parameter $\alpha$. Furthermore, previously
developed estimators achieve the minimax optimal rate for rather restricted
classes of sets (e.g., the boundary fragment and star-shaped sets) that
effectively reduce the set estimation problem to a function estimation problem.
This characterization precludes level sets with multiple connected components,
which are fundamental to many applications. This paper presents a fully
data-driven procedure that is adaptive to unknown regularity conditions and
achieves near minimax optimal Hausdorff error control for a class of density
level sets with very general shapes and multiple connected components.
