This paper deals with the density estimation on R d under sup-norm loss. We provide a fully data-driven estimation procedure and establish for it a so-called sup-norm oracle inequality. The proposed estimator allows us to take into account not only approximation properties of the underlying density, but eventual independence structure as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in the multi-variate density model. Usefulness of the developed approach is illustrated by application to adaptive estimation over anisotropic Nikolskii classes.