Any two arbitrary quantum states, acting on a given Hilbert space, need not be related by a majorization ordering, even if they are close to each other. Surprisingly, however, we prove that for any given state, and a ball of radius c around it (in trace distance) there exist two states in the ball, one which is majorized by all other states in it, and one which majorizes all other states in it. This result has diverse applications, e. g. , (i) in establishing continuity bounds for Schur concave functions, in particular for classes of entropies which arise naturally in quantum information theory, (ii) in approximate LOCC conversions of pure bipartite states, and (iii) in quantum state preparation. In obtaining the above result, we first obtain a more general result which might be of independent interest, namely a necessary and sufficient condition under which a state maximizes a concave and Gateaux-differentiable function in the ball. Examples of such a function include the von Neumann entropy, and the conditional entropy of bipartite states. Our proofs employ tools from the theory of convex optimization under non-differentiable constraints, in particular Fermat's Rule, and majorization theory. This is joint work with Eric Hanson.