Polynomial hulls and $H^\infty$ control for a hypoconvex constraint

Abstract.

We say that a subset of \({\mathbb C}^n \) is hypoconvex if its complement is the union of complex hyperplanes. Let \(\Delta \) be the closed unit disk in \({\mathbb C}\), \(\Gamma=\partial\Delta\). We prove two conjectures of Helton and Marshall. Let \(\rho \) be a smooth function on \(\Gamma\times{\mathbb C}^n\) whose sublevel sets have compact hypoconvex fibers over \(\Gamma\). Then, with some restrictions on \(\rho \), if Y is the set where \(\rho \) is less than or equal to 1, the polynomial convex hull of Y is the union of graphs of analytic vector valued functions with boundary in Y. Furthermore, we show that the infimum \(\inf_{f\in H^\infty(\Delta)^n}\|\rho(z,f(z))\|_\infty\) is attained by a unique bounded analytic f which in fact is also smooth on \(\Gamma\). We also prove that if \(\rho\) varies smoothly with respect to a parameter, so does the unique f just found.