In this work it is proved that, for the regular action of a semisimple irreducible algebraic group G on an affine space, the existence of a closed orbit of maximum dimension is equivalent to the existence of an invariant open set at any point of which the stationary subgroup is reductive. This result is established for the action of G on manifolds of a special type (the so-called factorial manifolds). There are given several other conditions equivalent to the existence of a closed orbit of maximum dimension for the action of G on an arbitrary affine manifold.