The convergence of hulls of curves

We prove results about limits of simple closed curves and polynomial hulls. The results show, in particular, that the polynomially convex, rectifiable simple closed curves in Cn, n ≥ 2, form a dense open subset in the space of all rectifiable simple closed curves with respect to both the supremum norm and the total variation norm. Also, under suitable hypotheses, the polynomial hull of the limit of a sequence of simple closed curves is the limit of the polynomial hulls.