In 2014, V.V. Andrievskii proved that the uniform norm of the nth-Chebyshev polynomial on a compact set K, consisting of a finite number of quasismooth arcs and Jordan domains with quasismooth boundary such that Ω:=C \K is a John domain, is bounded by the nth power of the logarithmic capacity of K. In this talk, we use the method of discretizing the equilibrium measure, due to V. Totik, to extend this result to unions of quasiconformal arcs and quasidisks.