In the present paper we characterize such convex closed sets in the real Hilbert space, that for each of these sets the operator of metric antiprojection on the set (which gives for a given point of the space the subset of points of the set, which are most farthest from the given point of the space) is singleton and Lipschitz continuous on the complementary to some neighborhood of the given set. We obtain new estimates of geometric properties of such a set as function of the size of the neighborhood of the set and the Lipschitz constant for the antiprojection operator. Properties of the metric antiprojector operator are investigated. We prove the stability of the antiprojection on a strongly convex set with respect to the point and to the set. We also consider the question: which points of the set are antiprojections of some points from the space?