  The paper deals with the problem of nonparametric estimating the $L_p$--norm,
$p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from
independent observations. The unknown density %to be estimated is assumed to
belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax
approach, and derive lower bounds on the minimax risk. In particular, we
demonstrate that accuracy of estimation procedures essentially depends on
whether $p$ is integer or not. Moreover, we develop a general technique for
derivation of lower bounds on the minimax risk in the problems of estimating
nonlinear functionals. The proposed technique is applicable for a broad class
of nonlinear functionals, and it is used for derivation of the lower bounds in
the~$L_p$--norm estimation.
