  We give a new proof for an equality of certain max-min and min-max
approximation problems involving normal matrices. The previously published
proofs of this equality apply tools from matrix theory, (analytic) optimization
theory and constrained convex optimization. Our proof uses a classical
characterization theorem from approximation theory and thus exploits the link
between the two approximation problems with normal matrices on the one hand and
approximation problems on compact sets in the complex plane on the other.
