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.