  The relative entropy of entanglement is defined in terms of the relative
entropy between an entangled state and its closest separable state (CSS). Given
a multipartite-state on the boundary of the set of separable states, we find a
closed formula for all the entangled state for which this state is a CSS. Quite
amazing, our formula holds for multipartite states in all dimensions. In
addition we show that if an entangled state is full rank, then its CSS is
unique. For the bipartite case of two qubits our formula reduce to the one
given in Phys. Rev. A 78, 032310 (2008).
