Chebyshev Polynomials on Compact Sets

In connection with a problem of H. Widom it is shown that if a compact set K on the complex plane contains a smooth Jordan arc on its outer boundary, then the minimal norm of monic polynomials of degree n = 1,2,... is at least (1 + β)cap(K)n with some β > 0, where cap(K)n would be the theoretical lower bound. It is also shown that the rate (1 + o(1))cap(K)n is possible only for compact for which the unbounded component of the complement is simply connected. A related result for sets lying on the real line is also proven.