**Abstract**
We study distance-like entanglement measures of multipartite states with certain symmetries. Using group averaging we show general conditions for which the relative entropy of entanglement, the geometric measure of entanglement and the logarithmic robustness are equal. We consider specific sets of states important in quantum information and many-body physics. We show equivalence of these measures for all stabilizer states, symmetric basis and antisymmetric basis states. We also calculate the explicit value of these measures for symmetric and antisymmetric basis states, indicating that antisymmetric states are generally more entangled. Finally we use inherent connections between these measures to entanglement witnesses to give classes of entanglement witnesses, for which the equivalence of these measures guarantees optimality of the witness.