Polynomially convex sets whose union has nontrivial hull

TL;DR

This paper investigates conditions under which pairs of polynomially convex sets in complex Euclidean spaces have unions that are not rationally convex, providing examples and approximation results for such sets.

Contribution

It demonstrates that polynomially convex sets can be embedded in complex spaces with unions that are not rationally convex and constructs specific examples including Cantor sets and arcs.

Findings

Existence of polynomially convex Cantor sets in C^3 with nonrationally convex union

Construction of disjoint polynomially convex sets with nonrationally convex union

Approximation of simple closed curves by locally polynomially convex curves

Abstract

Several results concerning pairs of polynomially convex sets whose union is not even rationally convex are given. It is shown that there is no restriction on how two spaces can be embedded in some $\C^N$ so as to be polynomially convex but have nonrationally convex union. It is shown that there exist two disjoint polynomially convex Cantor sets in $\C^3$ whose union is not rationally convex. The analogous assertion for arcs is also established. As an application it is shown that every simple closed curve in $\C^N$, $N\geq 3$, can be approximated uniformly by locally polynomially convex simple closed curves that are not rationally convex.