Chebyshev constants and the inheritance problem (Q1041630)

Denote by \(E\) a set of the form \(\bigcup_{j=1}^m [a_j,b_j]\) and let \(\mathcal{T}_n=\inf\|x^n+\dots\|_E\) be the \(n\)-th Chebyshev constant for \(E\). One of the main purposes of the paper is to give a new proof of the estimate \(\mathcal{T}_n \leq K \text{cap}(E)^n\), where \(K\) does not depend on \(n\). In addition, another result is given concerning the approximation of \(E\) by polynomial inverse images of \([-1,1]\) with order \(1/n\). The two theorems above are interrelated and arise from the new approach introduced by the author. This approach is based on the statement in the so-called inheritance problem, and has the advantage of avoiding the appearance of \(c\)-intervals, with the technical difficulties that they entail. A section is devoted to give another application of the latter problem in a more general setting.