The weighted Chebyshev polynomials are the monic polynomials that minimize the weighted sup-norm on a given set and monic orthogonal polynomials minimize the \(L^2\) norm associated with a Borel measure. We survey results involving the lower bounds of the norms of these extremal polynomials. We also discuss some recent results on asymptotics for the lower bounds, which generalize some classical results. In addition, we prove a new result regarding the lower bound for weighted Chebyshev polynomials which is valid on certain Cantor-type sets of zero Lebesgue measure.