  A scheme for locally adaptive bandwidth selection is proposed which
sensitively shrinks the bandwidth of a kernel estimator at lowest density
regions such as the support boundary which are unknown to the statistician. In
case of a H\"{o}lder continuous density, this locally minimax-optimal bandwidth
is shown to be smaller than the usual rate, even in case of homogeneous
smoothness. Some new type of risk bound with respect to a density-dependent
standardized loss of this estimator is established. This bound is fully
nonasymptotic and allows to deduce convergence rates at lowest density regions
that can be substantially faster than $n^{-1/2}$. It is complemented by a
weighted minimax lower bound which splits into two regimes depending on the
value of the density. The new estimator adapts into the second regime, and it
is shown that simultaneous adaptation into the fastest regime is not possible
in principle as long as the H\"{o}lder exponent is unknown. Consequences on
plug-in rules for support recovery are worked out in detail. In contrast to
those with classical density estimators, the plug-in rules based on the new
construction are minimax-optimal, up to some logarithmic factor.
