Asymptotic expressions of the form $({2 / \pi })\log n + c + r_n $ are investigated for the Lebesgue constants associated with interpolation at the Chebyshev nodes T and the “expanded Chebyshev nodes” $\hat T$. Estimations of the error $r_n $, are given. Similar asymptotic expressions can be obtained for interpolation at the Chebyshev extrema U and trigonometric interpolation at equidistant nodes. The deviation between the local maxima of the Lebesgue function is studied. It is shown that for the Chebyshev nodes T this deviation is less than $({2 / \pi })\log 2 = 0.441 \cdots $, whereas for $\hat T$ it is asymptotically not exceeding $({2 / \pi })\log 2 - {4 / {(3\pi ) = 0.016}} \cdots $.