  The geometric measure of entanglement, which expresses the minimum distance
to product states, has been generalized to distances to sets that remain
invariant under the stochastic reducibility relation. For each such set, an
associated entanglement monotone can be defined. The explicit analytical forms
of these measures are obtained for bipartite entangled states. Moreover, the
three qubit case is discussed and argued that the distance to the W states is a
new monotone.
