  In this paper we establish necessary and sufficient conditions for the
existence of line segments (or flats) in the sphere of the nuclear norm via the
notion of simultaneous polarization and a refined expression for the
subdifferential of the nuclear norm. This is then leveraged to provide
(point-based) necessary and sufficient conditions for uniqueness of solutions
for minimizing the nuclear norm over an affine manifold. We further establish
an alternative set of sufficient conditions for uniqueness, based on the
interplay of the subdifferential of the nuclear norm and the range of the
problem-defining linear operator. Finally, using convex duality, we show how to
transfer the uniqueness results for the original problem to a whole class of
nuclear norm-regularized minimization problems with a strictly convex fidelity
term.
