Chebyshev constants and the inheritance problem

It is proven that any set E consisting of finitely many intervals can be approximated with order 1/n by polynomial inverse images of [-1,1]. This leads to a new proof of the fact that the n-th Chebyshev constant is ⩽Kcap(E)n with some K independent of n. The proof uses properties of monotone systems, in particular, the statement in the so-called inheritance problem.