Polynomial hulls and H-infinity control
for a hypoconvex constraint.
Abstract
We say that a subset of Cn is hypoconvex if its
complement is the union of complex hyperplanes. Let D be the closed
unit disk in C, T= the boundary of D. 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× Cn --> 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, we show that the minimax
inff esssupz in T
|p(z,f(z))|
(where f may be bounded analytic on D) is attained
by a unique bounded analytic f which is also smooth on T. We also
prove that if p varies smoothly with respect to a parameter, so does
the unique f just found.