  The geometric measure of entanglement of a pure state, defined by its
distance to the set of pure separable states, is extended to multipartite mixed
states. We characterize the nearest disentangled mixed state to a given mixed
state with respect to this measure by a system of equations. The entanglement
eigenvalue for a mixed state is introduced. For a given mixed state, we show
that its nearest disentangled mixed state is associated with its entanglement
eigenvalue.
