  We study the estimation, in Lp-norm, of density functions defined on [0,1]^d.
We construct a new family of kernel density estimators that do not suffer from
the so-called boundary bias problem and we propose a data-driven procedure
based on the Goldenshluger and Lepski approach that jointly selects a kernel
and a bandwidth. We derive two estimators that satisfy oracle-type
inequalities. They are also proved to be adaptive over a scale of anisotropic
or isotropic Sobolev-Slobodetskii classes (which are particular cases of Besov
or Sobolev classical classes). The main interest of the isotropic procedure is
to obtain adaptive results without any restriction on the smoothness parameter.
