Geodesic
Non-Convexity in Best Complex Chebyshev Approximation by Rational Functions
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 115-116

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In real Chebyshev approximation by generalized rational functions, constraining denominators to be positive guarantees that the set of best coefficients is convex [1, 181]. We show by means of an example that denominators must be constrained to be of constant argument for such a convexity result to hold in complex Chebyshev approximation. 
Dunham, Charles B. Non-Convexity in Best Complex Chebyshev Approximation by Rational Functions. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 115-116. doi: 10.4153/CMB-1974-021-3
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[1] 1. Brosowski, B., Über die Eindeutigkeit der rationalen Tschebysheff-Approximationen, Numer. Math. 7 (1965), 176-186. Google Scholar

[2] 2. Dolganov, R. L., The approximation of continuous complex-valued functions by generalized rational functions, Siberian Math. J. 11 (1970), 932-942. Google Scholar

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