  Let $K$ be a infinite field, $S=K[x_1,\ldots,x_n]$ and $0\subset I\subsetneq
J\subset S$ two squarefree monomial ideals. In a previous paper we proved a new
formula for the Hilbert depth of $J/I$. In this paper, we illustrate how one
can use the Stanley-Reisner correspondence between (relative) simplicial
complexes and (quotients of) squarefree monomial ideals, in order to reobtain
some basic properties of the Hilbert depth. More precisely, we show that
$\operatorname{depth}(J/I)\leq \operatorname{hdepth}(J/I)\leq \dim(J/I)$. Also,
we show that $\operatorname{hdepth}(I)\geq \operatorname{hdepth}(S/I)+1$, if
$S/I$ is Cohen-Macaulay.
