  We address the problem of adaptive minimax density estimation on $\bR^d$ with
$\bL_p$--loss on the anisotropic Nikol'skii classes. We fully characterize
behavior of the minimax risk for different relationships between regularity
parameters and norm indexes in definitions of the functional class and of the
risk. In particular, we show that there are four different regimes with respect
to the behavior of the minimax risk. We develop a single estimator which is
(nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii
classes. Our estimation procedure is based on a data-driven selection of an
estimator from a fixed family of kernel estimators.
