Almost optimal polynomial approximation on convex sets in \(\mathbb {C}\)

For a function f, continuous on a compact convex set K, and analytic in its interior, we construct a sequence of almost optimal polynomials that converge with a geometric rate at points of analyticity of f. We also give an example of a non-convex K that does not have analytic boundary, but nevertheless permits geometric convergence of almost optimal approximants to the function f.