Chebyshev polynomials are not always optimal

We are concerned with the problem of finding the polynomial with minimal uniform norm on E among all polynomials of degree at most n and normalized to be 1 at c. Here, E is a given ellipse with both foci on the real axis and c is a given real point not contained in E. Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this work, we show that this is not true in general. Moreover, we derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.