Suppose a decision problem is invariant under the action of a group. We prove some general results on the essential completeness of the nonrandomized equivariant decision rules in the set of all equivariant rules under convexity conditions for possibily infinite dimensional decision spaces and on the risk-equivalence of both sets in a LYAPUNOV type setting. Furthermore, HUNT-STEIN type theorems on the existence of equivariant A-minimax rules are derived.