Connecting the generalized robustness and the geometric measure of entanglement

The main goal of this paper is to provide a connection between the generalized robustness of entanglement $\left(R_g\right)$ and the geometric measure of entanglement $\left(E_{GME}\right)$. 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 $R_g\left(\rho \right)$ 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 $E_{GME}$.