  In this paper we consider the problem of estimating $f$, the conditional
density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$
in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$
is a fixed point. We define two different procedures of estimation, the first
one using kernel rules, the second one inspired from projection methods. Both
adapted estimators are tuned by using the Goldenshluger and Lepski methodology.
After deriving lower bounds, we show that these procedures satisfy oracle
inequalities and are optimal from the minimax point of view on anisotropic
H{\"o}lder balls. Furthermore, our results allow us to measure precisely the
influence of $\mathrm{f}\_X(x)$ on rates of convergence, where $\mathrm{f}\_X$
is the density of $X$. Finally, some simulations illustrate the good behavior
of our tuned estimates in practice.
