  This paper presents a novel approach for pointwise estimation of multivariate
density functions on known domains of arbitrary dimensions using nonparametric
local polynomial estimators. Our method is highly flexible, as it applies to
both simple domains, such as open connected sets, and more complicated domains
that are not star-shaped around the point of estimation. This enables us to
handle domains with sharp concavities, holes, and local pinches, such as
polynomial sectors. Additionally, we introduce a data-driven selection rule
based on the general ideas of Goldenshluger and Lepski. Our results demonstrate
that the local polynomial estimators are minimax under a $L^2$ risk across a
wide range of H\"older-type functional classes. In the adaptive case, we
provide oracle inequalities and explicitly determine the convergence rate of
our statistical procedure. Simulations on polynomial sectors show that our
oracle estimates outperform those of the most popular alternative method, found
in the sparr package for the R software. Our statistical procedure is
implemented in an online R package which is readily accessible.
