We completely solve in the range space the feasibility problem and determine the structure of the slack set, in all three cases, namely strict tangent, internal, and external cases and then draw the consequences for the domain space. We show that in the range space the slack set is a g-polytope in the strictly tangent case and a flat bottom cup in all other cases. We conclude further developing in Hilbert spaces the theory of flat bottom cups, which we initiated in lcs.