  We use singular value decomposition to derive a tight lower bound for
geometric discord of arbitrary bipartite states. In a single shot this also
leads to an upper bound of measurement-induced non locality which in turn
yields that for Werner and isotropic states the two measures coincide. We also
emphasize that our lower bound is saturated for all $2\otimes n$ states. Using
this we show that both the generalized Greenberger-Horne-Zeilinger and $W$
states of $N$ qubits satisfy monogamy of geometric discord. Indeed, the same
holds for all $N$-qubit pure states which are equivalent to $W$ states under
stochastic local operations and classical communication. We show by giving an
example that not all pure states of four or higher qubits satisfy monogamy.
