The Lipschitz condition for the most farthest point in the Hilbert space

In the present work 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 a subset of points of the set, which are most farthest from the given point of the space) is singleton and coincides the Lipschitz condition on the complementary to some neighborhood of the given set. We obtain new estimates of geometric properties of such set as function of the size of the neighborhood of the set and the Lipschitz constant for the antiprojection operator.