The problem of finding the polynomial with minimal uniform norm on \({\mathcal E}_ r\) among all polynomials of degree at most \(n\) and normalized to be 1 at \(c\) is considered. Here \({\mathcal E}_ r\) is a given ellipse with both foci on the real axis and \(c\) is a given real point not contained in \({\mathcal E}_ r\). \textit{A. J. Clayton} [UKAEA Memorandum, AEEW, M 348 (1963)] proved that the unique optimal polynomial \(p_ n(z,r,c)\) is just the polynomial \((t_ n(z,c)=T_ n(z)/T_ n(c)\), where \(T_ n(z)\) denotes the \(n\)-th Chebyshev polynomial. The author shows that the normalized Chebyshev polynomials \(t_ n\) are not always optimal and hence Clayton's result is not true in general. Moreover, sufficient conditions which guarantee that Chebyshev polynomials are optimal are given.