  The main goal of this paper is to provide a connection between the
generalized robustness of entanglement ($R_g$) and the geometric measure of
entanglement ($E_{GME}$). First, we show that the generalized robustness is
always higher than or equal to the geometric measure. Then we find a tighter
lower bound to $R_g(\rho)$ based only on the purity of $\rho$ and its maximal
overlap to a separable state. As we will see it is also possible to express
this lower bound in terms of $E_{GME}$.
