Polynomial hulls and H-infinity control for a hypoconvex constraint

Abstract

We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and analytic selections,'' Indiana Univ. Math. J. 39, no. 1 (1990), 157-184.) Let p:T X C^n --> R+ be a smooth function whose sublevel sets have compact hypoconvex fibers over T. Then, with some restrictions on p, if Y is the set where p 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, let t be the smallest real number such that the set where p is less than or equal to t contains the boundary of the graph of some analytic vector-valued function on the disk. Then there is only one analytic vector-valued function f such that p(z,f(z)) is less than or equal to t for all z in T. We show that f is smooth on T. We also prove that if p varies smoothly with respect to a parameter, so does the unique f just found.