  In this paper, we give a sharp lower bound for the minimum deviation of the
Chebyshev polynomial on a compact subset of the real line in terms of the
corresponding logarithmic capacity. Especially if the set is the union of
several real intervals, together with a lower bound for the logarithmic
capacity derived recently by A.Yu.\,Solynin, one has a lower bound for the
minimum deviation in terms of elementary functions of the endpoints of the
intervals. In addition, analogous results for compact subsets of the unit
circle are given.
